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Calculus III (Spring 2024) |
Lecture Notes |
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1/18/2024: We discussed properties of real numbers. We defined addition, subtraction, and scalar mutiplication in $\mathbb{R}^n$. We derived the distance formula in $\mathbb{R}^n$ and graphed a couple of equations in $\mathbb{R}^3$. Notes from 1/18/24 |
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1/19/2024: We practiced graphing equations in $\mathbb{R}^3$. We also discussed the geometric interpretations and algebraic properties of vector addition and scalar multiplication in $\mathbb{R}^n$. Notes from 1/19/24 |
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1/23/2024: We worked a couple of applications of vector operations, and then we discussed the dot product. We discussed properties of the dot product, the geometric formula for dot product, the angle between vectors, orthogonality. We used the dot product to find equations of planes. Notes from 1/23/24 |
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1/25/2024: We discussed the Cauchy-Schwarz inequality, determinants of matrices and their geometric meaning, the cross product in R^3 and its properties. We used the cross product to find the equation of a plane in a particular example. Notes from 1/25/24 |
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1/26/2024: We worked more examples of utilizing dot products and cross products to calculate equations of lines and planes in $\mathbb{R}^3$. Notes from 1/26/24 |
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1/30/2024: We worked on graphing quadric surfaces. We also discussed parametrized curves in $\mathbb{R}^n$, as well as velocity, speed, acceleration. Notes from 1/30/24 |
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2/1/2024: We looked at properties of curves and vector-valued functions and their derivatives. We discussed the unit tangent vector and curvature. Notes from 2/1/24 |
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2/2/2024: We calculated tangential components of acceleration, calculated integrals of vector-valued functions, and thought about geometric meanings of equations in different dimensions. Notes from 2/2/24 |
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2/6/2024: We reviewed for the test on Thursday. Notes from 2/6/24 |
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2/9/2024: We discussed functions of two variables and contour diagrams. Notes from 2/9/24 |
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2/13/2024: We discussed domains and limits of functions of two variables, and we defined and calculated partial derivatives. Notes from 2/13/24 |
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2/15/2024: We discussed the meaning of partial derivatives, estimating them from contour diagrams, and the chain rule in several variables. Notes from 2/15/24 |
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2/16/2024: We discussed partial derivatives and directional derivatives. We mentioned the theorem that mixed partials commute. Notes from 2/16/24 |
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2/20/2024: We discussed the gradient, the differential, the derivative matrix, matrix operations, and the chain rule for the multivariable derivative. Notes from 2/20/24 |
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2/22/2024: We discussed meaning of the gradient and derivative matrix and used the gradient to calculate some tangent lines and tangent planes. Notes from 2/22/24 |
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2/23/2024: We found the critical points of a function of two variables, and we explored how to determine the type of critical point. We introduced the Hessian matrix. Notes from 2/23/24 |
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2/27/2024: We learned how to classify critical points of a real-valued function of several variables by using the eigenvalues of the Hessian matrix. Notes from 2/27/24 |
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2/29/2024: We discussed computing the Hessian and its eigenvalues at critical points of multivariable functions. We used sagemath to check the derivatives and solve the systems of algebraic equations. We also discussed the Lagrange multipliers method of finding critical points of a function of several variables when restricted to a curve/surface/hypersurface given as a level set of another function. Notes from 2/29/24 (includes completed example) |
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3/1/2024: We showed the contour diagram and ellipse from the homework question using sagemath plots. Also we stated the extreme value theorem. We found the maximum and minimum of a function on the unit disk, finding critical points in the interior and the ones on the boundary (using Lagrange multipliers). Notes from 3/1/24 sagemath code for the contour diagram sagemath code for the finding maxes/mins |
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3/7/2024: We used Lagrange multipliers to solve optimization problems. Notes from 3/7/24 |
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3/8/2024: We computed double integrals and discussed the meaning of what multidimensional integrals compute. Notes from 3/8/24 |
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3/19/2024: We used Lagrange multipliers to find distances between curves and surfaces. We also computed double integrals and changed the order of integration. Notes from 3/19/24 Example of using sagemath to compute multidimensional integrals |
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3/21/2024: We computed double integrals, changed orders of integration, and started discussing polar coordinates. Notes from 3/21/24 (with calculation finished) |
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3/22/2024: We computed double and triple integrals with and without polar coordinates. Notes from 3/22/24 |
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3/26/2024: We computed double and triple integrals with polar and cylindrical coordinates.
We defined spherical coordinates in $\mathbb{R}^3$. Notes from 3/26/24 |
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3/28/2024: We did a few more computations in polar and cylindrical coordinates, and we derived the volume form in spherical coordinates. We did one triple integral in spherical coordinates. Notes from 3/28/24 (includes completed calculation) |
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4/2/2024: We did a few more triple integral computations and also worked on changing coordinates in double integrals. Notes from 4/2/24 |
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4/4/2024: We learned how to calculate integrals of scalar functions over a curve and line integrals of a vector field over a curve. Notes from 4/4/24 |
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4/5/2024: We drew a couple of vector fields and calculated line integrals of vector fields that are conservative, using the fundamental theorem of calculus. Notes from 4/5/24 |
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4/9/2024: We reviewed a bit and then calculated a couple of line integrals, which can be written in either vector or differential form notation. Notes from 4/9/24 |
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4/12/2024: We discussed vector fields and determining potential functions from conservative vector fields. We learned fast ways to determine if a vector field is conservative in $\mathbb{R}^2$ and $\mathbb{R}^3$, using the curl of a vector field. Notes from 4/12/24 |
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4/16/2024: We computed curl and divergence of vector fields and differentials of one- and two-forms. We used Stokes'/Green's Theorem to turn a line integral into a two-dimensional area integral in the plane. Notes from 4/16/24 |
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4/18/2024: We discussed all the different versions of 1-dimensional, 2-dimensional, and 3-dimensional integrals, including those involving vector fields and differential forms, and the relations between them given by Stokes' theorem and the Fundamental Theorem of Calculus. Notes from 4/18/24 |
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4/19/2024: We discussed parametrizing surfaces and integrating scalar functions over surfaces. Notes from 4/19/24 |
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4/23/2024: We practiced calculating integrals of functions over surfaces and also converted them to line integrals using Stokes' Theorem. We also did differential form versions of the same integrals. Notes from 4/23/24 |
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4/25/2024: We worked on vector field flux integrals over surfaces and did examples of computing these over a closed surface by using Stokes' / Divergence Theorem. We also observed identities satisfied by curl, divergence, and gradient. Notes from 4/25/24 |
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4/26/2024: We utilized the Stokes' and Divergence Theorems to translate the integral forms of the Maxwell equations to differential equations. Notes from 4/26/24 |
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4/30/2024: We reviewed curve, surface, and space integrals and the Stokes and Divergence Theorems. Notes from 4/30/24 |
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