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

$\mathrm{D}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}\subseteq {\mathbb{ℝ}}^n$

Level Curves

Topographical map of a function

The value of $f(x,y)$ 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:

$f_x(x,y)=\frac{\partial f}{\partial x}(x,y)=\frac{\partial z}{\partial x}(x,y)$

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

$\begin{array}{rl}f(x,y)& =3x\sin (x^2+y^2)\\ f_x(x,y)& =3\sin (x^2+y^2)+3x\cos (x^2+y^2)\cdot x^2\\ f_y(x,y)& =3x\cos (x^2+y^2)\cdot 2y\end{array}$

Another Example

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

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