About
Statistics
Algebra
Trigonometry and Vectors
Calculus (1st Year)
Advanced Math
Vector Calculus, Linear Algebra and Differential
Equations
- Vector Valued Functions
- The Derivative of a Vector Valued Function
- Tangent Vectors
- Curvature
- Double Integrals in Polar Coordinates
- Triple Integrals
- Vector Fields
- Divergence
- Line Integrals
- Fundamental Theorem of Line Integrals
- Flux
- Surface Integrals
- Stokes Theorem
- The Divergence Theorem
- The Need for Linear Algebra
- Matrices
- Coordinate Transformations and Matrices
- Linear Combinations, Linear Independence, and Basis
- Linear Transformations
- The Magnitude of the Displacement Vector is Invariant Under the Rotation Matrix
- Vector Spaces
- Dimension of a Vector Space
- Eigenvalues and Eigenvectors
- General Differential Equations
- Separable Differential Equations
- Autonomous Differential Equations
- First Order Linear Differential Equations
- Integrating Factors
- Second Order Differential Equations Homogeneous With Constant Coefficients
- UC Functions
- Systems of Differential Equations
Course Topic: Vector Valued Functions
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-2
Time: 58:27 to 62:51
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video looks at two parallel oppositely charged plates and considers the motion of a particle that starts in the middle with a given initial velocity. The professor explains that the path of the particle is described by a vector valued function. The professor derives the equation for this path and then explains that this is how TV screens work. This is a relevant application of vector valued functions that students may not have seen before and could get them excited to learn this topic.
Course Topic: The Derivative of a Vector Valued Function
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-2
Time: 38:18 to 41:32
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the formula for the derivative of a two dimensional vector valued function. The professor gives a very clear mathematical derivation, but this clip has no physics applications contained in it. It would be good to show that what happens in a math class occurs in the same way in a physics class.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-2
Time: 49:05 to 51:26 (or to 52:29 to hear a physical application of this formula.)
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the derivative of the vector valued function that describes a particle moving at a constant velocity around a circle. Then the professor takes a second derivative to show that the acceleration (a(t) = r''(t)) is just a constant times the original vector valued function r(t). This is an important and simply derived application of derivatives of vector valued functions. He then reveals in a funny way that a(t) = v2/r.
Course Topic: Tangent Vectors
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-9
Time: 4:06 to 7:30
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video goes over the force by a magnetic field on a current moving along a semicircle. The integration is just sin, but there is a tougher derivation to arrive at the integrand that involves the cross product. of the tangent vector and the magnetic field. He uses the fact that the tangent vector is perpendicular to the position vector in this case. This is a complicated example, but the level is appropriate for second year calculus students.
Course Topic: Curvature
Time: 1:01:55 to 1:03:26
University: MIT
Course: Exploring Black Holes: General Relativity and Astrophysics
Professor Name: Edmund Bertschinger
Teaching Ideas: This video generalizes the curvature that is learned in calculus to curvature in four dimensional time-space. The math will be too difficult for the students to understand, but they will hear about curvature and how it is the extension of the curvature in calculus.
Video Link: https://www.youtube.com/watch?v=fIB5AE4SRN4&index=9&list=PL0A0E275BC354C934
Time: 17:03 to 21:19
University: Missouri University of Science and Technology
Course: Engineering Geology and Geotechnics
Professor Name: David Rogers
Teaching Ideas: This video explains that the thalweg (fast flowing part) of a stream occurs where the curvature is largest. The professor tell the story of how the railroad came to a halt on a curve on a mountain since the train will naturally slow down going uphill where the curvature is large. He does not do any mathematics, but it is clear from the picture that geologists and hydrologists need to perform this calculation. This is an excellent introduction to curvature, especially at a place where erosion control is important.
Course Topic: Double Integrals in Polar Coordinates
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-4
Time: 72:50 to 75:03
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video shows how to perform a double integral in polar coordinates to find the area of a triangle with vertices (0,0), (1,0), and (1,1). Although the professor just does the math without explaining how it applies to physics, the students get to see how a physicist approaches a double integral using slightly different notation.
Course Topic: Triple Integrals
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-12
Time: 28:26 - 35:47
University: Yale
Course: Freshman Organic Chemistry I
Professor Name: Michael McBride
Teaching Ideas: This video infers the application of triple integrals to finding the overlap density of two atoms' electron shells. The professors only talks about the integral and never states that it is a triple integral due to the fact that this is a freshman class; however, more mathematically sophisticated students will be able to see that the integral written down is a triple integral.
Time: 9:48 to 12:39
University: MIT
Course: Exploring Black Holes: General Relativity and Astrophysics
Professor Name: Edmund Bertschinger
Teaching Ideas: This video presents the gravitational potential function in as a triple integral over the density function divided by the magnitude of the displacement vector from the object to each point in the solid. The professor only presents the equation in this clip without doing any computations or examples.
Course Topic: Vector Fields
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-2
Time: 30:32 to 32:15
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video sketches the vector field that shows the electric field induced by a single charge. The professor is very clear about focusing on the magnitude and direction for this vector field that obeys the inverse square law. This is a great way to introduce vector fields to students.
Course Topic: Divergence
Video Link: https://www.youtube.com/watch?v=V5T_ywvofpE&list=PLE73AA240E8655D16&index=22
Time: 9:08 - 11:50
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video derives a quantum mechanical formula for the momentum operator in the radial direction. The derivation involves taking the divergence of a vector field using spherical coordinates (the professor uses r for the radial coordinate). The professor skips the steps of finding the divergence, so it would be a good exercise for the students to do this calculation to verify the professor's work. The physics will be far too difficult for the students to relate to, but some enjoy seeing high level work.
Course Topic: Line Integrals
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6
Time: 34:15 to 36:13
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the line integral in the context of comparing kinetic energy and work. The professor talks about cutting up a curve into tiny pieces and adding up all the work components. This will be an nice reinforcement and application when first introducing line integrals.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6
Time: 47:50 to 51:52
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video shows how to evaluate a line integral to find work by going along a path y = x2. It makes use of the fact that y is a function of x and does not talk about parameterized curves. The first few seconds summarizes another path, so this video emphasizes that the choice is path can be important. This video nicely goes over two concepts in a short time and can serve as a tie between what is done in calculus and physics.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-23
Time: 20:48 to 23:22
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the work done by a gas in going through a cycle that begins along an isotherm continues along a constant pressure line and then up the constant volume line to get where one started. The professor shows that this is a line integral and stresses the fact that going around the other direction results in the opposite work done. This is a nice clip when talking about orientation of a curve.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5
Time: 14:56 to 17:24
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the formula that the change in kinetic energy is equal to the total work done, and thus the conservation of energy theorem in three dimensions. The derivation uses derivatives of vector valued functions and line integrals. It is easy to understand and can serve as a powerful example to illustrate the use of the line integral.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5
Time: 21:14 to 24:19
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video gives an example of a force that is not independent of path. The professor goes through the line integral calculations along both paths. It is a simple example, but it demonstrates the process of finding line integrals well.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-8
Time: 4:24 to 6:17
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video explains what the EMF is for an electric circuit. The professor derives it using a line integral, but it is intuitive enough that he does not have to parameterized the curve and integrate. This is more of a motivating example of why line integrals are important than an example of how to work out a line integral.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-9
Time: 51:35 to 54:55
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the Ampere's Law, which gives a formula for the magnetic field along circular field lines a fixed distance from a wire with a current. The professor takes the line integral along a circle deriving the integrand for this which eventually just becomes 1, so the line integral is just the length of the curve times the constant that the professor pulled out in the derivation. This is an important and simple use of the line integral.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-10
Time: 31:22 to 32:58
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video calculates the EMF induced by a square wire as it moves horizontally. By definition, it is a line integral. The professor calculates it by observation since the integrands are either 0 or other constants. This is an easy to understand use of the line integral.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-14
Time: 29:15 to 29:55
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video explains that the sum of the line integrals over two curve that have a common edge is equal to the line integral over the union of the curves with the common edge deleted. The professor does not show an application in this clip, but the explanation is clear.
Course Topic: Fundamental Theorem of Line Integrals
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6
Time: 61:51 to 63:53
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video explains how the fundamental theorem of line integrals in two dimensions works. The professor does not name the theorem specifically, but he does check that the appropriate partials are equal. This is a clear example of the FTLI in physics.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6
Time: 63:55 to 64:51
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video shows that gravity is a conservative vector field and thus satisfies the conservation of energy law. This is an example where although the language of math and physics may differ, the fundamentals are the same. The example is very easy since the vector field is a constant, but the importance in physics is clear. To see the concept applied to a rollercoaster go until 67:57.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6
Time: 69:47 to 70:26
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video presents a challenge for the students where the instructor posits that if the force is parallel to the position then it is conservative. Since the professor just asks the question without presenting the solution, an instructor can ask the students to solve the problem in class. Having students do an application right out of a physics class will show them that solving math problems is more than just an exercise.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5
Time: 25:06 to 26:44
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video shows that independence of path for line integrals is equivalent to the statement that the line integral over any closed curve equals zero. This is identical to how it is introduced in a vector calculus class, but it can be motivating to see it happen in a physics class.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5
Time: 27:16 to 31:15
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video presents the statement of the fundamental theorem of line integrals. The statement is the same as it is in a vector calculus class even though the professor is presenting it in the context of physics. This will serve as a reinforcement of this important theorem.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5
Time: 45:05 to 47:05
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video demonstrates the fundamental theorem of line integrals applies to gravity. This is a very basic example, but shows that the FTLI applies to something as important as gravity.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5
Time: 55:30 to 59:12
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video uses the FTLI by selecting a convenient path to find the potential energy difference from one point to another through an electric field. This serves as a solid application of the FTLI and goes through the process of integrating well. The professor does make an error and fixes it, but that will not confuse the students.
Time: 48:38 to 50:01
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video explains that enthalpy is independent of path. The professor defines this as a "State Function". Since this is just a first year chemistry course, she does not get into any vector calculus explanations, but does have a friendly diagram of a mountain to use an an analogy.
Course Topic: Flux
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-12
Time: 35:06 - 37:59
University: Yale
Course: The Atmosphere, the Ocean, and Environmental Change
Professor Name: Ronald Smith
Teaching Ideas: This video defines the mass flux and the heat flux through a cylindrical conduit. In the second part, the professor explains a slide that shows that there is heat flux while there is no mass flux. It only defines it for constant vector fields and is not very high level, but it can be used as a motivator when teaching flux in a multivariate calculus course.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-11
Time: 44:52 to 46:05
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video considers two coils and looks at the EMF based on the flux through the coils. The professor reminds the students about the definition of the magnetic flux and then explains why coiling a wire give a much stronger EMF. This is a relevant application of the flux.
Course Topic: Surface Integrals
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-3
Time: 57:46 to 60:14
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video explains Gauss' Law that states that the flux through any closed surface containing a charge q is equal to the charge over ε0. This is a nice introduction to flux integrals. The professor describes it geometrically which explains the idea of what a surface integral is.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-4
Time: 33:27 to 34:36
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video looks at the flux through a sphere with a charge inside. The professor explains that since the direction is outward, the cos is1 and the field strength is a constant along the sphere. Thus the integrand is just a constant giving the constant times the area of the sphere. This is a nice example that shows the idea of the flux integral.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-9
Time: 60:49 to 62:35
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video goes over four fundamental equations, two of which involve line integrals and two involve surface integrals. The equation come for electric and magnetic fields over closed curves and closed surfaces. No calculus is done, but is is nice to see both line integrals and surface integrals compared.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-11
Time: 15:54 to 19:45
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video uses a surface integral and other calculus techniques to find the momentum of as a function of the average magnetic field. The calculation of the surface integral uses an average value theorem that is analogous to the average value theorem for single integrals. This video contains a nice combination of calculus techniques and works well to summarize much of what is done in the full calculus sequence.
Course Topic: Stoke's Theorem
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-10
Time: 46:40 to 48:47
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video relates the change in the magnetic flux and the electrical work done, called Faraday-Lenz' Law. The result is that the change in flux through the surface is equal to the line integral on the boundary of the electric force. This is very similar to Stoke's Theorem, although the proof will be too difficult for the students, it can be a nice introduction to Stoke's Theorem.
Course Topic: The Divergence Theorem
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-3
Time: 62:26 to 64:23
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video relates Gauss' Law to a triple integral. It will not be obvious from the video, but this basically shows that Gauss' Law is a result of the divergence theorem. Students may need some more reference for this and an article that explains it is at: http://www2.ph.ed.ac.uk/~rhorsley/SII08-09_mp2a/lec20_2.pdf (See the part on Source and Sinks).
Course Topic: The Need for Linear Algebra
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-22
Time: 42:38 to 43:58
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video talks about the investigation of when a function can be written as a linear combination of other functions. The professor also states that the solution is unique. The professor states, "If you learn linear algebra one day, it is all linear algebra. This is an excellent video to show on the first day of a linear algebra course, since the professor advertises linear algebra so well.
Course Topic: Matrices
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-33
Time: 11:19 - 16:12
University: Yale
Course: The Nature of Evolution: Selection, Inheritance and History
Professor Name: Stephen C. Stearns
Teaching Ideas: This video explains basic game theory and payoff matrices using having a hawk vs. dove attitude. The professor goes into detail on how it works and defines what it means to be a stable strategy. At the end he asks the students to work out what strategy is best and why. In class students at this point can also discuss with each other the best strategy.
Time: 45:35 - 48:32
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video explains that Gaussian elimination is used to solve the system of equations that are produced from analyzing a circuit. The equations are shown, but the matrix and solution is not. The professor just explains that it can be done.
Time: 56:21to 58:37
University: MIT
Course: Exploring Black Holes: General Relativity and Astrophysics
Professor Name: Edmund Bertschinger
Teaching Ideas: This video describes Einstein's tensor which is a 4x4 matrix that includes the Lorenz factor. This video will be over the heads of the students, but is a nice exploration into the rigor or upper division physics courses.
Course Topic: Coordinate Transformations and Matrices
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-13
Time: 9:12 to 10:10
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video writes down the picture and the equations of the coordinate transformation that corresponds to rotation of the plane by an angle θ. The professor does not write down the corresponding matrix, but linear algebra students can easily be asked to write down this matrix. The instructor will need to tell the students that the professor is providing an analogy to the Lorenz transformations that occur in Einstein's theory of relativity.
Course Topic: Linear Combinations, Linear Independence, and Basis
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-21
Time: 29:30 to 35:55
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video remarks that any vector can be written uniquely as a linear combination of of the basis vectors. Then the professor explains how to find the coefficients in the case that the basis is orthonormal. The purpose of this is to find the Fourier coefficients of a given functions, but students do not need to understand Fourier Analysis in order to understand the video.
Video Link: https://www.youtube.com/watch?v=aoaR0BsBR2U&list=PLE73AA240E8655D16&index=2
Time: 26:30 - 28:22
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video looks the complete set of states formed by spin + and spin - of an electron. The professor explains that these two states form a basis for the complete set of states. He then compares this with the three dimensional real vector space. This is a great example that compares what is done in linear algebra with what is done in quantum mechanics.
Video Link: https://www.youtube.com/watch?v=qCwUKsWt8z4&list=PLE73AA240E8655D16&index=3
Time: 18:59 - 20:38
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video explains the physical nature of the basis vectors of the energy of a particle. This give somewhat tangible meaning to the idea of basis vectors. Instead of just abstract ideas, these basis vectors are actual energy levels. This gives a somewhat concrete example of the abstract concept of basis vectors.
Time: 8:57 to 13:06
University: MIT
Course: Artificial Intelligence
Professor Name: Patrick Winston
Teaching Ideas: This video explains how computers recognize manufactured objects. The professor explains that if you select four point on the four transformed pictures then there is a unique transformation that will identify an object once one object's picture is taken from three different perspectives. The professor does not say so, but this works because the points form a basis for the vector space. This is a simplified version of how the self driving car can differentiate between a car and a bike for example. This clip can also be shown in a college algebra class, but the students will understand it in a simpler way.
Course Topic: Linear Transformations
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-13
Time: 13:28 to 13:07
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video points out that the Lorenz transformations that occur in Einstein's theory of relativity are linear transformations. The instructor can tell the students that in order to understand Einstein's theories, one must first have a firm grounding in linear transformations and linear algebra in general.
Time: 13:50 - 17:57
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video defines the linear transformation using the vocabulary of physics. The professor uses the term "homogeneity" for L(kv) = kL(v) and "superposition" for L(v+w) = L(v) + L(w). This is a nice way of seeing the two properties of linear transformations in use.
Video Link: https://www.youtube.com/watch?v=aoaR0BsBR2U&list=PLE73AA240E8655D16&index=2
Time: 14:30 - 17:30
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video looks bras and kets and the natural way of defining a basis for the bras so that they are the canonical linear transformations. The professor proves that this gives the sum of the amplitudes which must add up to 1. This is a great way of showing students how linear transformations are used in high level physics. They will follow some, but not all of the explanation, so it is important to tell the students that they are not responsible for it on the exam.
Course Topic: The Magnitude of the Displacement Vector is Invariant Under the Rotation Matrix
Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-12
Time: 4:30 - 6:40
University: Yale
Course: Frontiers and Controversies in Astrophysics
Professor Name: Charles Bailyn
Teaching Ideas: This video looks at the invariant effect of the rotation matrix on the displacement vector. For the next few minutes he discusses how this also works for the space-time metric tau which is also invariant under 4x4 matrix rotations. This can be looked at when discussing the generalized inner product of a space and how the main application is to the metric of space-time.
Course Topic: Vector Spaces
Video Link: https://www.youtube.com/watch?v=aoaR0BsBR2U&list=PLE73AA240E8655D16&index=2
Time: 2:27 - 6:12
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video looks at the ket notation for quantum mechanics and based on the definitions the professor concludes that they form a vector space (over the complex numbers). This is a great example of vector spaces looking completely different from what students are familiar with: directed line segments. The students will need to be told that the ket is used to describe particles in quantum mechanics, but the field is too difficult to get into any further details.
Video Link: https://www.youtube.com/watch?v=aoaR0BsBR2U&list=PLE73AA240E8655D16&index=2
Time: 9:00 -12:09
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video looks at the bra notation for quantum mechanics and based on the definitions the professor concludes that they form a vector space (over the complex numbers). This is another great example of vector spaces looking completely different from what students are familiar with: directed line segments. It also always gets a chuckle based on its name. This is the beginning of a very deep field that the students will not be expected to understand at this point in their studies.
Course Topic: Dimension of a Vector Space
Video Link: https://www.youtube.com/watch?v=aoaR0BsBR2U&list=PLE73AA240E8655D16&index=2
Time: 12:16 - 14:10
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video looks at the dimension of the space of bras and states that it is the same as the dimension of the space of kets. The professor does this by starting with a basis ket vector, define a bra that is 1 on that ket and 0 on the rest of the ket basis vectors. This is a simple proof using very difficult vector spaces. Show this video to demonstrate that linear algebra extends beyond the vectors that students are familiar with.
Course Topic: Eigenvalues and Eigenvectors
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-26
Time: 19:00 - 20:34
University: Yale
Course: The Nature of Evolution: Selection, Inheritance and History
Professor Name: Stephen C. Stearns
Teaching Ideas: This video introduces the idea of a Leslie matrix to come up with the population in the next generation given the current generation's adult and child populations and their rates. The professor just refers to the fact that a matrix can be constructed and linear algebra can be used, but does not show the matrix. It is recommended that after presenting this clip, the class is show another website that actually gives the Leslie matrix. One such is here: http://en.wikipedia.org/wiki/Leslie_matrix. This is a nice application that clearly shows the use of matrices outside of mathematics. A class can further use eigenvalues and eigenvectors to show that the dominant eigenvalue is the long term growth rate and the eigenvector is the stable age distribution.
Video Link: https://www.youtube.com/watch?v=qCwUKsWt8z4&list=PLE73AA240E8655D16&index=3
Time: 46:48 - 48:55
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video derives the eigenkets and eigenbras which are analogous to eigenvectors and eigenvalues. This uses some very familiar mathematics to understand some very difficult physics, quantum mechanics. This will be fascinating for the students who are going into physics but will be meaningless to those who do not know physics.
Video Link: https://www.youtube.com/watch?v=cVbB6wFNqYc&index=4&list=PLE73AA240E8655D16
Time: 6:40 - 7:56
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video shows that the matrix of a linear transformation with respect to the basis of eigenvectors will be diagonal with eigenvalues along the diagonal. The professor uses the "ket" language to describe the situation, but it is the same mathematics.
Video Link: https://www.youtube.com/watch?v=G7G1mjx8mlQ&list=PLE73AA240E8655D16&index=23
Time: 22:15 - 26:27
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video derives the equations for the spin angular momentum. The professor finds the eigenvector for the corresponding matrix. He uses the same techniques that are used in linear algebra. The math is quite complex, using the half angle formulas and the physics is even more complex. This video can be used only to impress students about the complex work done in advanced physics.
Course Topic: General Differential Equations
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-22
Time: 39:14 to 41:42
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video writes down and briefly displays the Schrodinger Equation. The professor does not solve it, but does briefly explain what it does. This video can be shown at the beginning of a differential equations class to show what differential equations look like and how they are used.
Course Topic: Separable Differential Equations
Video Link: https://www.youtube.com/watch?v=IgaTSenVDwI&list=PL48DE756A5800ED5F&index=5
Time: 30:05 to 32:57
University: UC Berkeley
Course: Environmental Science
Professor Name: (Not Provided in Video)
Teaching Ideas: This video explains why the Allee Effect which is the same as logistics model with threshold. The professor explains why this can result in harvesting can drive species to extinction. She does not solve any differential equations, but the graphs tell a compelling story of why taking into account threshold can prevent a species from being driven to extinction.
Course Topic: Autonomous Differential Equations
Video Link: https://www.youtube.com/watch?v=xOkksGIBoWY&list=PL48DE756A5800ED5F&index=7
Time: 22:06 to 24:33
University: UC Berkeley
Course: Environmental Science
Professor Name: (Not Provided in Video)
Teaching Ideas: This video explains the Levin's model of population that takes into account metapopulations or populations in slightly connected patches. The professor show how this has to do with the extinction rate of each patch and the ability of species to migrate from one patch to another. The differential equation is presented and the graph of the dependent variable vs. the derivative is shown. She also talks about stable equilibrium.
Course Topic: First Order Linear Differential Equations
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-8
Time: 18:48 to 21:03
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This videoStarts with the equation that corresponds with an RC circuit. This equation is a first order linear separable differential equation. The professor splits up the problem by writing the solution as the limiting solution plus the rest. Although this is not the traditional mathematical way that this type of equation is solved, it demonstrates that there are many techniques to simplify differential equations that make them easier to solve.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-12
Time: 7:50 to 8:52
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video sets up the equation for an LR circuit. In this clip, the differential equation is not solved, but only set up. It is not a difficult equation to solve, so students can be asked to solve it in class given that L, R, and V0 are constants. If you continue to 13:00 the solution is shown using the same technique as the previous clip.
Course Topic: Integrating Factors
Video Link: https://www.youtube.com/watch?v=pjFYiBEp1Ug&list=PLE73AA240E8655D16&index=8
Time: 22:10 - 25:44
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video goes through some of the mathematics behind deriving the equations that give the stationary state for the harmonic oscillator. The professor explicitly goes over the integrating factor approach to solving first order linear differential equations. The math matches up well with what is done in a standard differential equations class, but the physics will be way over the heads of sophomore level students. This will just serve as a reminder that even though their freshmen level physics does not use complicated techniques to solving differential equations, that does not mean that they won't have to use it in their upper level classes.
Video Link: https://www.youtube.com/watch?v=KTAp0Q0GZhE&index=21&list=PLE73AA240E8655D16
Time: 3:39 - 7:13
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video derives the quantum mechanical angular momentum operator L+ on ψll. The equation is a first order linear differential equation that the professor solves using the method of integrating factors. The physics will be too difficult for the students to relate to, but some enjoy seeing high level work. The math done is at just the right level for differential equation students.
Video Link: https://www.youtube.com/watch?v=s1At6bW8UX0&list=PLE73AA240E8655D16&index=26
Time: 5:46 - 8:56
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video derives the equations for the states of the hydrogen atom. The derivation uses the method of integrating factors. The instructor will have to explain that "the ground state" means the lowest energy state. Students who have had chemistry will recognize this as the 1s shell for the electron.
Course Topic: Second Order Differential Equations Homogeneous With Constant Coefficients
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-3
Time: 47:30 to 48:25
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the differential equation -kx = mx'' which gives the position of a mass on a spring. The professor states, "then you go to the math department and say please tell me what's the answer to this equation." At this point the instructor can tell the class that they are the math department and they must tell the professor the answer.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-17
Time: 18:19 to 24:19
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video solves the second order differential equation x'' + ω02x = 0 using the technique of substituting y = Aeαt into the differential equation in order to come up with simple harmonic motion.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-17
Time: 25:16 - 29:00
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video shows that the harmonic oscillator is a linear differential equation, thus any linear combination of solutions gives a solution. This is easy to follow, but the physics is not apparent in the explanation.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-17
Time: 39:00 43:44
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video solves the differential equation that corresponds to a harmonic oscillator with friction. The professor uses the standard technique that is used in a differential equations class, so this can replace the derivation that occurs in the differential equations lecture on solving second order homogeneous differential equations with constant coefficients. The instructor should note that the professor made a mistake in calling the solution -α where it should have been α. The professor later notices the mistake, but spends an unreasonable time correcting it. Instead of showing this error fixing clip that comes next, the instructor can just continue it and demonstrate the cases. This can easily replace the lecture on this material starting with the professor's clip and finishing with the instructor's comments about damped, critically damped, and over damped systems.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-12
Time: 25:06 to 29:34
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the differential equation for an LC circuit. The professor notes that it has the same exact form as the mass spring system and uses the fact that he knows the solution of the spring mass system to solve it. This gives two fundamental applications in the same example and can work as a great introduction to second order differential equations.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-12
Time: 30:52 to 32:28
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video demonstrates with the LC circuit example that any linear combination of two solutions to a linear differential equation is also a solution to the differential equation. This will help students remember this essential fact about solutions.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-12
Time: 43:37 to 46:14
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video looks at the LRC circuit example with no forcing function. The professor does not derive the solution, but just write down the answer and compares it with the projectile motion with friction. This can be done by the instructor after going over the theory. The professor gives the general idea for solving.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-22
Time: 43:58 to 47:19
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video solve the Schrodinger equation for the very special case where the middle term vanishes and the final is just a constant. This is a particle moving around a ring with no potential energy. The professor uses complex powers, so this can be shown in the section on complex roots. At the beginning of the clip, the professor states, "Every great discovery in physics is accompanied by some mathematical stuff you need." This is a perfect motivator for math students.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-22
Time: 65:36 to 68:58
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video solves the Schrodinger equation a particle in a well. The clip looks at just the part to the right of the well and uses the standard method of assuming exponential form. The roots are real and distinct. This is an easy to understand example of solving second order homogeneous differential equations.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-22
Time: 71:29 to 75:35
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video solves the Schrodinger equation a particle in a well. The clips starts just after the general solution is found and focuses on finding the constants. The derivation finishes with one of the greatest discoveries of physics: that quantized nature of particles. The math is pretty easy to follow and the result is clear.
Time: 28:38 - 33:42
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video uses the step by step process in finding the solution to a second order homogeneous linear differential equation. It doesn't explicitly use UC functions, but rather just emphasizes the fact that the general solution is the particular solution plus the homogeneous solution. The professor is very clear and thorough in his explanation. The circuit is not shown in this clip, so the instructor will want to explain that this is for an LC circuit.
Time: 45:36 - 49:10
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video goes over a heuristic method of solving LRC circuits that can be described by second order linear homogeneous differential equations. The professor uses the language and of circuit analysis to sketch the solution. This is a nice supplement that shows how differential equation mixes with practical work.
Course Topic: UC Functions
Time: 21:11 - 23:45
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video uses the UC function "A" to solve an LRC circuit. He goes through the process in 4 steps. The first 2 1/2 minutes involve describing the steps. Then if there is time, the next five minutes solve the problem. This nicely mirrors what is done in a differential equations class, but uses the LRC circuit as the example.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-17
Time: 60:36 to 66:53
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video solves the physics problem of an object subject to a spring like force, friction, and a forcing function that is a cos function. the professor uses a trick involving a combination of complex numbers and UC functions to solve it without using any trig functions. The professor explains how to obtain the real part in the next seven minutes. This shows an alternate method of solving such differential equations.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-13
Time: 26:22 to 30:13
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video explains that after finding the particular solution to a second order differential equation, one must then add in the homogeneous solution. The context is with LRC circuits, but the explanation applies to any differential equation. This is a very clear explanation of why this addition is required and it explains the transient vs. steady state current.
Time: 35:08 to 43:43
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video explains the step by step process in finding the solution to a non-homogeneous linear differential equation. It does it for one of the simplest examples that is just first order and the forcing function is just a constant. It doesn't explicitly use UC functions, but rather just emphasizes the fact that the general solution is the particular solution plus the homogeneous solution. The professor is very clear and thorough in his explanation. The example is for solving a circuit, but the explanation will work for any non-homogeneous linear differential equation.
Time: 28:38 - 34:45 (or 39:27 to view the solving of the constants given the initial conditions)
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video uses the step by step process in finding the solution to a second orderhomogeneous linear differential equation. It doesn't explicitly use UC functions, but rather just emphasizes the fact that the general solution is the particular solution plus the homogeneous solution. The professor is very clear and thorough in his explanation. The circuit is not shown in this clip, so the instructor will want to explain that this is for an LC circuit.
Time: 35:21 - 39:28
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video compares the method of using the UC function Asint + Bcost with the alternative approach Aet. The professor calls the second the "Sneaky" or "Shore" method and using the analogy of the choice between going down class 5 rapids vs. walking along that part of the shore and getting back in after the rapids. This is a comedic approach to the rough concept and will help the students remember the concepts.
Course Topic: System of Differential Equations
Video Link: https://www.youtube.com/watch?v=K0wrnwWRE00&index=10&list=PL48DE756A5800ED5F
Time: 8:36 to 14:33
University: UC Berkeley
Course: Environmental Science
Professor Name: (Not Provided in Video)
Teaching Ideas: This video goes over the basics of the Latka Voltera equations that describe the population model for two species in competition for the same resource. The professor just goes over why the differential equations are reasonable. He does not solve the equations. In the second half of this clip, he shows three examples, two where one of the species is driven to extinction and the other where the two species end up in a stable equilibrium with both species making it. If you don't want to play it for a full six minutes, you can stop at 11:34 and just see the differential equations and not the examples.
Video Link: https://www.youtube.com/watch?v=Eb-EIwv6LBM&index=11&list=PL48DE756A5800ED5F
Time: 32:04 to 37:13
University: UC Berkeley
Course: Environmental Science
Professor Name: (Not Provided in Video)
Teaching Ideas: This video also looks at the Latka Voltera equations that describe the population model for two species in competition, but in this case the predator prey relationship. The professor explains how the components of the differential equations are formed, but she does not solve the equations. She does show a graph and explains the cyclic nature of the solution. At the last few seconds of the clip, she show that the model works for the snowshoe hare and the lynx.