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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is even a variable, pretend it's a constant, and differentiate normally with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .

Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align}}

Another Example

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial z}{\partial x}} using "implicit differentiation".

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}