In preparation for multivariate calculus, we revise basic concepts in 1 variable calculus, including functions, limits, and continuity.
In preparation for multivariate calculus, we revise integration techniques on one variable functions.
In this multivariate calculus tutorial we revised basic concepts from one variable calculus in preparation for multivariate calculus. This includes derivates, integrals, definite integrals, sketching functions, evaluating functions at a point, limits, continuity, and differentiability. We will also consider functions of two variables that are not continuous at a particular point.
In preparation for multivariate calculus, we do some exercises in geometric thinking. This includes problems with parabolas, hyperbolas, and ellipses.
In this lecture we will learn about partial derivatives, their meaning, how to calculate partial derivatives, and how to use them.
In this lecture we go deeper into the meaning of partial derivatives using Mathematica animations and consider some applications.
We study implicit differentiation, which allows the calculation of dy/dx for a point on a implicitly defined curve such as x^2+y^2 = 1, not just a point on the explicit curve y = f(x). We use Mathematica to visualize curves and surfaces in 3D where we are calculating derivatives implicitly.
In this multivariate calculus tutorial we learn about the meaning of partial derivatives, calculate partial derivatives both via the limit definition and by using rules for differentiation. We differentiate functions partially using the product rule, the quotient rule, and the chain rule. Next we calculate the derivatives of implicitly defined functions, curves and surfaces. Finally, we calculate second partial derivatives and verify Clairaut's theorem with some examples.
We cover some additional background in preparation for more multivariate calculus by working through some exercises in geometric thinking.
We discuss mathematical notation, piecewise functions, integration, inverse functions, and Mathematica.
The gradient vector and directional derivative are covered. Several Mathematica plots illustrate these. We finish the lecture with global maxima and minima problems.
In this lecture we study Lagrange multipliers to find the max or min of a function subject to a constraint.
In this lecture we explain how Lagrange multipliers work.
In this multivariate calculus tutorial we calculate the gradient vector, the directional derivative, find and classify critical points identifying maxima and minima, and use Lagrange multipliers to find the maxima and minima of a function subject to a constraint.
We resume geometric thinking exercises in preparation for multivariate calculus problems, covering vector and scalar equations of the plane.
We learn about the divergence and curl of vector functions with some applications including work done and conservative vector fields. Illustrations made with Mathematica are shown.
We calculate work done of a vector field over a linear path, use the curl vector, and give an example of Green's theorem.
In this multivariate calculus tutorial we work through problems on grad(f), div(F), and curl(F), where F(x, y, z) is a vector field and f(x, y, z) is a scalar function. We then do problems applying these concepts including showing that a vector field is conservative and finding a potential function f such that F = grad f. We conclude with an example of Green's theorem in the plane and Stokes' theorem to illustrate the use of the curl vector.
We resume the geometric thinking exercises and discuss lines in 3D, the vector equation of a line to ready you for considering parametrization of lines for multivariate calculus problems.
We study double integrals and triple integrals, swapping the order of integration with Fubini's theorem and we learn to swap the order of integration when the region is not rectangular also. Several geometric visualizations are made using Mathematica.
In this lecture we will study double integrals with a change of variables to polar coordinates and other coordinate systems. This includes calculation of the Jacobian matrix.
In this multivariate calculus tutorial we work through problems on double integrals, Fubini's theorem, changing the order of integration, volume under a surface, and finish with flux calculations including an example of the divergence theorem.
In this lecture we study parametrization of lines and curves for the purpose of learning to calculate line integrals. This includes arc length calculations. In this discussion we provide Mathematica code for displaying various graphics associated with line integrals.
We study the fundamental theorem of line integrals which gives us an easy way to calculate line integrals. This means that there is an easy way to calculate work done by a vector field from point A to point B when that vector field is conservative since it will be path independent and we can use the fundamental theorem. We give several examples and show Mathematica code for displaying the associated graphics.
In this multivariate calculus tutorial we work through problems on parametrizing curves so that we can calculate line integrals. We then calculate work done and finish with an application of the fundamental theorem for line integrals.
In this lecture we do a multivariate calculus exam in 42 minutes. This exam covers partial derivatives, double and triple integrals, directional derivatives, grad(f), div(F), curl(F), double integrals, maximum rate of change of a multivariate function, conservative vector fields, potential functions, work done, parametrization of a path, integration by substitution, line integrals, the fundamental theorem for line integrals, the divergence theorem, surface integrals, the divergence theorem, and Green's theorem.
Here is a series of practical exercises in Mathematica which will help you calculate and visualize your work. These are pdf files, so you will have to type out the code yourself.
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