MATH 251 Lecture 8

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Domains

$n$-dimensional points that ensure that the result of a function will be a real number

$\mathrm {Domain} \subseteq \mathbb {R} ^{n}$

Level Curves

Topographical map of a function

The value of $f\left(x,\,y\right)$ is constant on a level curve (contour)

Level Surfaces

Same as Level Curves, but for 4-dimensional objects. For example, areas of equal potential or temperature in space.

Partial Derivatives

Notation:

$f_{x}\left(x,\,y\right)={\frac {\partial f}{\partial x}}\left(x,\,y\right)={\frac {\partial z}{\partial x}}\left(x,\,y\right)$

In the partial derivative above, forget that $y$ is even a variable, pretend it's a constant, and differentiate normally with respect to $x$.

Example

${\begin{aligned}f\left(x,\,y\right)&=3x\sin {\left(x^{2}+y^{2}\right)}\\f_{x}\left(x,\,y\right)&=3\sin {\left(x^{2}+y^{2}\right)}+3x\cos {\left(x^{2}+y^{2}\right)}\cdot x^{2}\\f_{y}\left(x,\,y\right)&=3x\cos {\left(x^{2}+y^{2}\right)}\cdot 2y\end{aligned}}$

Another Example

Find ${\frac {\partial z}{\partial x}}$ using "implicit differentiation".

${\begin{aligned}\ln {\left(3\,x^{2}+y^{2}-z^{2}\right)}&=10\\{\frac {\partial }{\partial x}}\left[\ln {\left(3\,x^{2}+y^{2}-z^{2}\right)}\right]&={\frac {1}{3x^{2}+y^{2}-z}}\left(6x-{\frac {\partial z}{\partial x}}\right)\\{\frac {\partial }{\partial x}}\left[10\right]&=0\\0&={\frac {6\,x-z_{x}}{3\,x^{2}+y^{2}-z}}\\0&=6x-z_{x}\\z_{x}&=6x\end{aligned}}$