Multivariate Calculus

Abigail Huettig

5 Multivariate Calculus

Multivariate calculus is the study of applying calculus concepts to functions of more than one independent variable. For example, when we had one independent variable, we found

$\lim_{x\to c} f(x).$

With two, we find

$\lim_{(x,y)\to (c,d)} f(x,y).$

Instead of finding what $f(x)$ approaches as $x$ gets closer to $c$ , we find what $f(x,y)$ gets closer to as $x$ gets closer to $c$ and $y$ gets closer to $d$ . Additionally, the limit must be the same when $(c,d)$ is approached from any direction in order for the limit to exist, just as the left and right limits had to be equal with one independent variable [15]. Regarding continuity, $f(x,y)$ is continuous at a point $(c,d)$ when

$\lim_{(x,y)\to (c,d)} f(x,y)=f(c,d).$

Finding limits and continuity of three or more independent variables is done similarly.

We can also integrate and differentiate functions with more than one independent variable. We will not be discussing integration, but we will discuss partial differentiation.

Partial Differentiation

Partial differentiation is what we do to find partial derivatives. But what is a partial derivative? When we did differentiation before, we found $\frac{d}{dx} f(x)$ . That is, we found the derivative of $f(x)$ with respect to its independent variable $x$ . But with multivariate calculus, we have functions with more than one independent variable. We cannot find the derivative of $f(x,y)$ with respect to $x$ and $y$ at the same time. So instead, we find the partial derivative of $f(x,y)$ with respect to $x$ and the partial derivative $f(x,y)$ with respect to $y$ . Larson and Edwards provide us with the formal definition [15].

Definition I.11
If $x=f(x,y)$ , then the first partial derivatives of $f$ with respect to $x$ and $y$ are the functions $f_x$ and $f_y$ defined by

$f_x (x,y)=\lim_{\Delta x\to 0} \frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}$

and

$f_y (x,y)=\lim_{\Delta y\to 0} \frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}$

provided the limits exist.

What this is saying is that the partial with respect to $x$ is finding the derivative of $f(x,y)$ by acting as if $x$ is the only independent variable. Finding the partial with respect to $y$ is done in a similar manner.

It is easy to see how these ideas can be applied to functions with more than two independent variables. To find the partial with respect to some variable, we treat that variable as the only one and hold the others constant.

The notation is a little different as well. Larson and Edwards present us with a variety of ways that a partial can be denoted [15]. The partial of $z=f(x,y)$ with respect to $x$ looks like

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

The partial with respect to $y$ or any other independent variable is written in a similar way. Let’s now look at a problem I completed for Calculus III that comes from Larson and Edwards [15].

Example 13
Find the first partial derivatives of $w=\displaystyle\frac{7xz}{x+y}$ with respect to $x$ , $y$ , and $z$ .

Solution
Using the quotient rule, for $w_x$ , we have

$\begin{align*} w_x&=\frac{(x+y)\frac{dw}{dx}(7xz)-(7xz)\frac{dw}{dx}(x+y)}{(x+y)^2} \\ &=\frac{(x+y)(7z)\frac{dw}{dx}(x)-(7xz)\frac{dw}{dx}(x+y)}{(x+y)^2} \\ &=\frac{(x+y)(7z)(1)-(7xz)(1)}{(x+y)^2} \\ &=\frac{7xz+7yz-7xz}{(x+y)^2} \\ &=\frac{7yz}{(x+y)^2} \\ \text{}\\ w_y&=7xz\frac{dw}{dy}[(x+y)^{-1}] \\ &=7xz[-1(x+y)^{-2}](1) \\ &=-\frac{7xz}{(x+y)^2} \\ \text{}\\ w_z&=\frac{7x}{x+y}\frac{dw}{dz}(z) \\ &=\frac{7x}{x+y}(1) \\ &=\frac{7x}{x+y} \end{align*}$

We are now going to switch gears and discuss sequences and series. At first glance, these topics seem like they do not belong with the other topics we have discussed thus far. However, we will see that limits largely influence what we discuss relating to sequences and series.

5 Multivariate Calculus

Partial Differentiation

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