forked from eggy/eifueo
111 lines
4.1 KiB
Markdown
111 lines
4.1 KiB
Markdown
# MATH 119: Calculus 2
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## Multivariable functions
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!!! definition
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- A **multivariable function** accepts more than one independent variable, e.g., $f(x, y)$.
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The signature of multivariable functions is indicated in the form *[identifier]*: *[input type]* → *[return type]*. Where $n$ is the number of inputs:
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$$f: \mathbb R^n \to \mathbb R$$
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!!! example
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The following function is in the form $f: \mathbb R^2\to\mathbb R$ and maps two variables into one called $z$ via function $f$.
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$$(x,y)\longmapsto z=f(x,y)$$
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### Sketching multivariable functions
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!!! definition
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- In a **scalar field**, each point in space is assigned a number. For example, topography or altitude maps are scalar fields.
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- A **level curve** is a slice of a three-dimensional graph by setting to a general variable $f(x, y)=k$. It is effectively a series of contour plots set in a three-dimensional plane.
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- A **contour plot** is a graph obtained by substituting a constant for $k$ in a level curve.
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Please see [level set](https://en.wikipedia.org/wiki/Level_set) and [contour line](https://en.wikipedia.org/wiki/Contour_line) for example images.
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In order to create a sketch for a multivariable function, this site does not have enough pictures so you should watch a YouTube video.
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!!! example
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For the function $z=x^2+y^2$:
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For each $x, y, z$:
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- Set $k$ equal to the variable and substitute it into the equation
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- Sketch a two-dimensional graph with constant values of $k$ (e.g., $k=-2, -1, 0, 1, 2$) using the other two variables as axes
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Combine the three **contour plots** in a three-dimensional plane to form the full sketch.
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A **hyperbola** is formed when the difference between two points is constant. Where $r$ is the x-intercept:
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$$x^2-y^2=r^2$$
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<img src="/resources/images/hyperbola.svg" width=600 />
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If $r^2$ is negative, the hyperbola is is bounded by functions of $x$, instead.
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## Limits of two-variable functions
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A function is continuous at $(x, y)$ if and only if all possible lines through $(x, y)$ have the same limit. Or, where $L$ is a constant:
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$$\text{continuous}\iff \lim_{(x, y)\to(x_0, y_0)}f(x, y) = L$$
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In practice, this means that if any two paths result in different limits, the limit is undefined. Substituting $x|y=0$ or $y=mx$ or $x=my$ are common solutions.
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!!! example
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For the function $\lim_{(x, y)\to (0,0)}\frac{x^2}{x^2+y^2}$:
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Along $y=0$:
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$$\lim_{(x,0)\to(0, 0)} ... = 1$$
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Along $x=0$:
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$$\lim_{(0, y)\to(0, 0)} ... = 0$$
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Therefore the limit does not exist.
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## Partial derivatives
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Partial derivatives have multiple different symbols that all mean the same thing:
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$$\frac{\partial f}{\partial x}=\partial_x f=f_x$$
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For two-input-variable equations, setting one of the input variables to a constant will return the derivative of the slice at that constant.
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By definition, the **partial** derivative of $f$ with respect to $x$ (in the x-direction) at point $(a, B)$ is:
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$$\frac{\partial f}{\partial x}(a, B)=\lim_{h\to 0}\frac{f(a+h, B)-f(a, B)}{h}$$
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Effectively:
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- if finding $f_x$, $y$ should be treated as constant.
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- if finding $f_y$, $x$ should be treated as constant.
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!!! example
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With the function $f(x,y)=x^2\sqrt{y}+\cos\pi y$:
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\begin{align*}
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f_x(1,1)&=\lim_{h\to 0}\frac{f(1+h,1)-f(1,1)} h \\
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\tag*{$f(1,1)=1+\cos\pi=0$}&=\lim_{h\to 0}\frac{(1+h)^2-1} h \\
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&=\lim_{h\to 0}\frac{h^2+2h} h \\
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&= 2 \\
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\end{align*}
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### Higher order derivatives
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!!! definition
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- **wrt.** is short for "with respect to".
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$$\frac{\partial^2f}{\partial x^2}=\partial_{xx}f=f_{xx}$$
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Derivatives of different variables can be combined:
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$$f_{xy}=\frac{\partial}{\partial y}\frac{\partial f}{\partial x}=\frac{\partial^2 f}{\partial xy}$$
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The order of the variables matter: $f_{xy}$ is the derivative of f wrt. x *and then* wrt. y.
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**Clairaut's theorem** states that if $f_x, f_y$, and $f_{xy}$ all exist near $(a, b)$ and $f_{yx}$ is continuous **at** $(a,b)$, $f_{yx}(a,b)=f_{x,y}(a,b)$ and exists.
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!!! warning
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In multivariable calculus, **differentiability does not imply continuity**.
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