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MATH 119: Calculus 2
Multivariable functions
!!! definition - A multivariable function accepts more than one independent variable, e.g., \(f(x, y)\).
The signature of multivariable functions is indicated in the form [identifier]: [input type] → [return type]. Where \(n\) is the number of inputs:
\[f: \mathbb R^n \to \mathbb R\]
!!! example The following function is in the form \(f: \mathbb R^2\to\mathbb R\) and maps two variables into one called \(z\) via function \(f\).
$$(x,y)\longmapsto z=f(x,y)$$Sketching multivariable functions
!!! definition - In a scalar field, each point in space is assigned a number. For example, topography or altitude maps are scalar fields. - 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. - A contour plot is a graph obtained by substituting a constant for \(k\) in a level curve.
Please see level set and contour line for example images.
In order to create a sketch for a multivariable function, this site does not have enough pictures so you should watch a YouTube video.
!!! example For the function \(z=x^2+y^2\):
For each $x, y, z$:
- Set $k$ equal to the variable and substitute it into the equation
- 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
Combine the three **contour plots** in a three-dimensional plane to form the full sketch.A hyperbola is formed when the difference between two points is constant. Where \(r\) is the x-intercept:
\[x^2-y^2=r^2\]
If \(r^2\) is negative, the hyperbola is is bounded by functions of \(x\), instead.
Limits of two-variable functions
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:
\[\text{continuous}\iff \lim_{(x, y)\to(x_0, y_0)}f(x, y) = L\]
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.
!!! example For the function \(\lim_{(x, y)\to (0,0)}\frac{x^2}{x^2+y^2}\):
Along $y=0$:
$$\lim_{(x,0)\to(0, 0)} ... = 1$$
Along $x=0$:
$$\lim_{(0, y)\to(0, 0)} ... = 0$$
Therefore the limit does not exist.