Lecture Notes
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Notes
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Video
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Notes from Aug 24:
We discussed vectors and vector operations.
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Video from Aug 24
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Notes from Aug 26:
We discussed the dot product and the properties of vector operations, and we applied our results to
finding equations of planes.
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Video from Aug 26
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Notes from Aug 27:
We discussed linear maps in higher dimensions using matrices, determinants, and the
cross product of vectors in R3.
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Video from Aug 27
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Notes from Aug 31:
We used vector operations to find equations of planes and did a little bit of 3-d graphing.
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Video from Aug 31
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Notes from Sep 2:
We did some examples of graphing equations in 3-d by using sections.
We also began to discuss parametrizations of curves.
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Video from Sep 2
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Notes from Sep 3:
We reviewed properties of vectors, dot product, cross product, determinant, and
we thought about some examples.
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Video from Sep 2
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Notes from Sep 7:
We found velocity and acceleration of curves.
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Video from Sep 7
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Notes from Sep 9:
We learned about hyperbolic trig functions.
We discussed velocity, speed, acceleration, arclength, unit tangent vectors, and
curvature of curves. We started talking about partial derivatives of
functions of more than one variable.
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Video from Sep 9
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Notes from Sep 10:
We discussed functions of several variables and partial derivatives.
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Video from Sep 10
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Notes from Sep 14:
We discussed partial derivatives, contour diagrams, and introduced the gradient vector field.
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Video from Sep 14
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Notes from Sep 16:
We discussed projecting one vector onto another, tangential component of acceleration of curves,
and gradients and contour plots of functions of several variables. We also did some computations
with sagemath.
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Video from Sep 16
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Notes from Sep 17:
We discussed properties of the gradient vector field and contours of functions of several variables.
We showed how to take partial derivatives and to make various plots with sagemath.
Link to sagemath code that we used.
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Video from Sep 17
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Notes from Sep 21:
We reviewed partial derivatives and contour diagrams.
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Video from Sep 21
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Notes from Sep 24:
We discussed matrix operations and the derivative matrix of functions of several
variables.
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Video from Sep 24
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Notes from Sep 28:
We discussed the derivative matrix and linear approximations to functions.
We used it to show that the gradient of a real-valued function is perpendicular to
its contours.
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Video from Sep 28
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Notes from Sep 30:
We discussed finding points of a surface where the tangent plane is parallel to a given plane.
We also introduced the concept of directional derivative.
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Video from Sep 30
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Notes from Oct 1:
We computed the directional derivative of some functions of several variables.
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Video from Oct 1
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Notes from Oct 5:
We did another example with computing directional derivatives. Then we
used the gradient to find critical points of a function.
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Video from Oct 5
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Notes from Oct 7:
We discussed examples of finding critical points. We also learned about eigenvalues
of matrices and about the Hessian matrix.
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no video from Oct 7 (oops)
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Notes from Oct 8:
We discussed a little about contour diagrams and gradients and directional derivatives.
Then we discussed the Hessian matrix and how to use its eigenvalues to determine the
type of a critical point for a function of several variables.
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Video from Oct 8
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Notes from Oct 12:
We discussed homework on critical points and directional derivatives, and we used the
Hessian to calculate the types of critical points in an example. We also discussed the
strategy for finding global maxima and minima for functions of several variables.
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Video from Oct 12
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Notes from Oct 21
sagemath code used in class
We showed how to compute critical points and Hessian eigenvalues using sagemath.
We also started working an optimization example, where we desired to find the absolute
maximum and minimum of a function of (x,y) on a region bounded by two curves.
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Video from Oct 21
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Notes from Oct 26
Worked on some optimization questions, finding the global max and min of functions
on different domains.
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Video from Oct 26
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Notes from Oct 28
Sagemath code used in class
We discussed using Sagemath to solve optimization questions, and we discussed the method of LaGrange multipliers.
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Video from Oct 28
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Notes from Oct 29
Sagemath code used in class
We discussed using Sagemath to solve Lagrange multiplier equations, and we discussed applications of optimization.
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Video from Oct 29
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Notes from Nov 2
Sagemath code used in class
We did some examples of optimization using Lagrange multipliers, including one example with two constraints.
We used sagemath to solve the nonlinear system of equations.
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Video from Nov 2
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Notes from Nov 4
We discussed the meaning of one-variable integrals and the fundamental theorem of calculus.
Then we generalized them to the case of definite integrals of functions of two variables over a region in the plane.
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Video from Nov 4
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Notes from Nov 5
We reviewed the Lagrange multiplier calculations from the homework, and we finished the
double integral we worked on Thursday.
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Video from Nov 5
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Notes from Nov 9
Sagemath code used in class
We discussed homework and review exercises. We also discussed double integrals starting
with either vertical or horizontal slices.
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Video from Nov 9
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Notes from Nov 12
We discussed substitution in integrals and thinking in terms of differential forms.
We showed the differential of a function of two variables.
We did another example of changing the order of integration.
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Video from Nov 12
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Notes from Nov 16
We discussed doing double integrals in polar coordinates, changing order of integration,
interpretations of single, double, and triple integrals as areas, volumes, and average
values. We did an example of setting up a triple integral, and we started an example of
computing a triple integral.
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Video from Nov 16
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Notes from Nov 18
sagemath used in class
We discussed techniques of integrating to find volumes and to find average values of functions of 3 variables.
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Video from Nov 18
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Notes from Nov 19
We reviewed parametrizations of curves and explored how to integrate a function over a
parametrized curve.
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Video from Nov 19
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Notes from Nov 30
We did some integrals of functions over curves and line integrals of vector fields over curves.
We discussed the Fundamental Theorem of Calculus for Line Integrals and conservative
vector fields.
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Video from Nov 30
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Notes from Dec 2
We discussed curl and divergence of vector fields, and we discussed how to think of
1-forms and 2-forms in R^3 as vector fields. We also introduced Green's and Stokes' theorems.
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Video from Dec 2
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Notes from Dec 3
We discussed line integrals, surface integrals, and Stokes' theorem.
We applied what we have learned to questions 3.15 and 3.21.
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Video from Dec 3
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Notes from Dec 7
We discussed surface integrals and the divergence theorem.
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Video from Dec 7
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