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Parametrized Surfaces Just as you describe curves in the plane parametrically with a pair of equations $x=f(t), y=g(t)$ defined on some parameter interval $I,$ you can sometimes describe surfaces in space with a triple of equations $x=f(u, v), y=g(u, v), z=h(u, v)$ defined on some parameter rectangle $a \leq u \leq b, c \leq v \leq d .$ Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section 16.5.) Use a CAS to plot the surfaces. Also plot several level curves in the $x y$ -plane.$$\begin{array}{l}x=2 \cos u \cos v, \quad y=2 \cos u \sin v, \quad z=2 \sin u \\0 \leq u \leq 2 \pi, \quad 0 \leq v \leq \pi\end{array}$$

$$0 \leq v \leq 2 \pi$$

Calculus 3

Chapter 14

Partial Derivatives

Section 1

Functions of Several Variables

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Parametrized Surfaces Just…

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Just as you describe curve…

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