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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.

Partial derivatives

Partial derivatives have multiple different symbols that all mean the same thing:

\[\frac{\partial f}{\partial x}=\partial_x f=f_x\]

For two-input-variable equations, setting one of the input variables to a constant will return the derivative of the slice at that constant.

By definition, the partial derivative of \(f\) with respect to \(x\) (in the x-direction) at point \((a, B)\) is:

\[\frac{\partial f}{\partial x}(a, B)=\lim_{h\to 0}\frac{f(a+h, B)-f(a, B)}{h}\]

Effectively:

• if finding \(f_x\), \(y\) should be treated as constant.
• if finding \(f_y\), \(x\) should be treated as constant.

!!! example With the function \(f(x,y)=x^2\sqrt{y}+\cos\pi y\):

\begin{align*}
f_x(1,1)&=\lim_{h\to 0}\frac{f(1+h,1)-f(1,1)} h \\
\tag*{$f(1,1)=1+\cos\pi=0$}&=\lim_{h\to 0}\frac{(1+h)^2-1} h \\
&=\lim_{h\to 0}\frac{h^2+2h} h \\
&= 2 \\
\end{align*}

Higher order derivatives

!!! definition - wrt. is short for “with respect to”.

\[\frac{\partial^2f}{\partial x^2}=\partial_{xx}f=f_{xx}\]

Derivatives of different variables can be combined:

\[f_{xy}=\frac{\partial}{\partial y}\frac{\partial f}{\partial x}=\frac{\partial^2 f}{\partial xy}\]

The order of the variables matter: \(f_{xy}\) is the derivative of f wrt. x and then wrt. y.

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.

!!! warning In multivariable calculus, differentiability does not imply continuity.

Linear approximations

A tangent plane represents all possible partial derivatives at a point of a function.

For two-dimensional functions, the differential could be used to extrapolate points ahead or behind a point on a curve.

\[ \Delta f=f'(a)\Delta d \\ \boxed{y=f(a)+f'(a)(x-a)} \]

The equations of the two unit direction vectors in \(x\) and \(y\) can be used to find the normal of the tangent plane:

\[ \vec n=\vec d_1\times\vec d_2 \\ \begin{bmatrix}-f_x(a,b) \\ -f_y(a,b) \\ 1\end{bmatrix} = \begin{bmatrix}1\\0\\f_x(a,b)\end{bmatrix} \begin{bmatrix}0\\1\\f_y(a,b)\end{bmatrix} \]

Therefore, the general expression of a plane is equivalent to:

\[ z=C+A(x-a)+B(x-b) \\ \boxed{z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)} \]

??? tip “Proof” The general formula for a plane is \(c_1(x-a)+c_2(y-b)+c_3(z-c)=0\).

If $y$ is constant such that $y=b$:

$$z=C+A(x-a)$$

which must represent in the x-direction as an equation in the form $y=b+mx$. It follows that $A=f_x(a,b)$. A similar concept exists for $f_y(a,b)$.

If both $x=a$ and $y=b$ are constant:

$$z=C$$

where $C$ must be the $z$-point.

Usually, functions can be approximated via the tangent at \(x=a\).

\[f(x)\simeq L(x)\]

!!! warning Approximations are less accurate the stronger the curve and the farther the point is away from \(f(a,b)\). A greater \(|f''(a)|\) indicates a stronger curve.

!!! example Given the function \(f(x,y)=\ln(\sqrt[3]{x}+\sqrt[4]{y}-1)\), \(f(1.03, 0.98)\) can be linearly approximated.

$$
L(x=1.03, y=0.98)=f(1,1)=f_x(1,1)(x-1)+f_y(1,1)(y-1) \\
f(1.03,0.98)\simeq L(1.03,0.98)=0.005
$$

Differentials

Linear approximations can be used with the help of differentials. Please see MATH 117#Differentials for more information.

\(\Delta f\) can be assumed to be equivalent to \(df\).

\[\Delta f=f_x(a,b)\Delta x+f_y(a,b)\Delta y\]

Alternatively, it can be expanded in Leibniz notation in the form of a total differential:

\[df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy\]

??? tip “Proof” The general formula for a plane in three dimensions can be expressed as a tangent plane if the differential is small enough:

$$f(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(x-b)$$

As $\Delta f=f(x,y)-f(a,b)$, $\Delta x=x-a$, and $\Delta y=y-b$, it can be assumed that $\Delta x=dx,\Delta y=dy, \Delta f\simeq df$.

$$\boxed{\Delta f\simeq df=f_x(a,b)dx+f_y(a,b)dy}$$

Related rates

Please see SL Math - Analysis and Approaches 1 for more information.

!!! example For the gas law \(pV=nRT\), if \(T\) increases by 1% and \(V\) increases by 3%:

\begin{align*}
pV&=nRT \\
\ln p&=\ln nR + \ln T - \ln V \\
\tag{multiply both sides by $d$}\frac{d}{dp}\ln p(dp)&=0 + \frac{d}{dT}\ln T(dt)-\frac{d}{dV}\ln V(dV) \\
\frac{dp}{p} &=\frac{dT}{T}-\frac{dV}{V} \\
&=0.01-0.03 \\
&=-2\%
\end{align*}

Parametric curves

Because of the existence of the parameter \(t\), these expressions have some advantages over scalar equations:

• the direction of \(x\) and \(y\) can be determined as \(t\) increases, and
• the rate of change of \(x\) and \(y\) relative to \(t\) as well as each other is clearer

\[ \begin{align*} f(x,y,z)&=\begin{bmatrix}x(t) \\ y(t) \\ z(t)\end{bmatrix} \\ &=(x(t), y(t), z(t)) \end{align*} \]

The derivative of a parametric function is equal to the vector sum of the derivative of its components:

\[\frac{df}{dt}=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2+\left(\frac{dz}{dt}\right)^2}\]

Sometimes, the chain rule for multivariable functions creates a new branch in a tree for each independent variable.

For two-variable functions, if \(z=f(x,y)\):

\[\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}\]

Sample tree diagram:

(Source: LibreTexts)

!!! example This can be extended for multiple functions — for the function \(z=f(x,y)\), where \(x=g(u,v)\) and \(y=h(u,v)\):

<img src="/resources/images/many-var-tree.jpg" width=300>(Source: LibreTexts)</img>

Determining the partial derivatives with respect to $u$ or $v$ can be done by only following the branches that end with those terms.

$$
\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial u} \\
$$

!!! warning If the function only depends on one variable, \(\frac{d}{dx}\) is used. Multivariable functions must use \(\frac{\partial}{\partial x}\) to treat the other variables as constant.