Multivariable Calculus
Level: 0 1 2 3 4 5 6 7
MSC Classification: msc26 (Real functions)

Prerequisites: Calculus I, Vectors and Space

Getting Oriented

Rough Guides to Analysis
Real Analysis
Complex Analysis

Multivariable calculus extends the notions of single-variable calculus to functions with more than one input and/or output.

The Basics

Multivariable Functions

Single-variable functions are typically represented in the form f(x), where x is the input (the independent variable, and f(x) is the output. One sometimes writes y=f(x), and then y is called the dependent variable. These functions are usually depicted visually by the graph, which formally is described as the collection of points $\{(x,f(x)) : x \in D$ over a given domain $D\subset\mathbb{R}$. The graph is a set of points in the Cartesian plane.

Multivariable calculus looks at functions of more than one variable. The functions are written in the form f(x,y), or more generally f(x1, x2, …, xn). Both x and y are independent variables. The graph of a multivariable function is the collection of points $\{(x,y,f(x,y)) : (x,y) \in D\}$, where $D\subset\mathbb{R}^2$ is the domain of the function. The domain is a subset of the plane whose points are pairs (x,y). The graph typically forms a surface, and the z value above the point (x,y) is the value of the function z=f(x,y).

Another way to represent such a function is a contour plot. A contour of a multivariable function is the curve represented by an equation f(x,y)=C for some constant C. (This is also called a level curve.) Contour plots put together several level curves in a single plot in the Cartesian plane.

In general, graphs of multivariable functions with n inputs exist in Rn+1, and contour plots exist in Rn. When n is greater than 2, the contours are generally called level surfaces.

Slices of Multivariable Functions

When either x or y is "held constant", one obtains a function with a single input. For example, the function $f(x,y)=x y^2$ has two inputs. For y=y0, it reduces to the function $f(x)=x y_0^2$. When y is held fixed in this manner, we will usually write $f(x,y_0)$, where the subscript 0 indicates that y is being held fixed, or we might write $f(x,3)$ if y0=3. The key point here is that, when one of the variables is fixed, the function can be interpreted as a function of a single variable. This is called a slice or trace of the function. Geometrically, fixing y corresponds to intersecting the graph z=f(x,y) with the plane y=y0. Analogous definitions hold in the case where x is fixed to x0.

One can go even further by looking at the values of the function f over a particular parametric curve $\langle x(t), y(t) \rangle$. Substituting these values in for x and y gives the function $f(x(t), y(t))$. Geometrically, this is the part of the graph directly above the curve $\langle x(t), y(t) \rangle$ in the xy-plane. For example, if $f(x,y)=x y^2$ and the curve is $\langle \cos(t), \sin(t) \rangle$ (representing the unit circle), then one has the function $f(x(t), y(t)) = \cos(t) \sin^2(t)$. Although this is a function of a single variable and could be plotted as such, the direct geometric interpretation is that $\cos(t) \sin^2(t)$ represents the value of the function f above the point $\langle x(t), y(t) \rangle = \langle \cos(t), \sin(t) \rangle$ in the domain.

Multivariable Differentiation

Rates of Change

Generically, the rate of change refers to how much the function changes as the input changes. For a single variable function f(x), this leads to the slope formula $\frac{\Delta f}{\Delta x}$.

Given a function f(x,y) of more than one variable, there are several ways to look at the function's "rate of change", since there is more than one input. Are both x and y changing? Are they changing in some related manner? Is just one of them changing? There are mathematical concepts for each of these ideas.

If just x is changing and y=y0 is fixed, then one writes $\frac{\Delta f}{\Delta x}\Big|_{y=y_0}$. If just x is changing and x=x0 is fixed, then one writes $\frac{\Delta f}{\Delta y}\Big|_{x=x_0}$. If both x and y are changing, rate-of-change concepts include the directional derivative and the gradient. These are somewhat more complex, but may be understood in terms of the "basic" rates $\frac{\Delta f}{\Delta x}$ and $\frac{\Delta f}{\Delta y}$.

Partial Derivatives

The derivative of a single variable function f(x) can be understood as the limit of secant slopes:

\begin{align} f'(x)=\frac{df}{dx}(x)=\lim_{\Delta x\to 0} \frac{\Delta f}{\Delta x} = \lim_{\Delta x\to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}. \end{align}

The definition of multivariable derivatives is essentially the same. The difference lies in the choice of which variable is changing.


Given a function f(x,y), the partial derivative of f with respect to x is $\lim_{\Delta x\to 0} \frac{\Delta f}{\Delta x}$, provided f is defined at (x,y) and the limit exists. Explicitly, $\Delta f$ measures the difference between the function at the initial point $f(x,y)$ and a nearby point $f(x+\Delta x,y)$, so the explicit limit being evaluated is

\begin{align} f_x(x,y)=\frac{\partial f}{\partial x}(x,y)=\lim_{\Delta x\to 0} \frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}. \end{align}

Both notations fx and $\frac{\partial f}{\partial x}$ are commonly used to represent the partial derivative.

In a similar fashion, one has the partial derivative of f with respect to y:

\begin{align} f_y(x,y)=\frac{\partial f}{\partial y}(x,y)=\lim_{\Delta y\to 0} \frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}. \end{align}

Other Derivative Concepts

The directional derivative is the concept capturing the idea of a slope, or rate of change, in a direction other than along the x or y axes.


Given a function f(x,y), and a direction $\hat{\mathbf{u}}$ (a unit vector), the directional derivative of f in the direction of u is

\begin{align} D_{\hat{\mathbf{u}}} f(x_0,y_0) = \lim_{\Delta t\to 0} \frac{f(x+u_1\Delta t,y+u_2\Delta t) - f(x)}{\Delta t}, \end{align}

where $\hat{\mathbf{u}}=\langle u_1,u_2\rangle$.

Frequently, problems are stated which ask for the directional derivative without specifying a unit vector. The approach is then to find the unit vector in the same direction that is specified. Here are two specific cases:
  • The directional derivative for a non-unit vector v is $D_{\hat{\mathbf{u}}} f$ where $\hat{\mathbf{u}}}=\frac{\mathbf{v}}{||\mathbf{v}||}$.
  • The directional derivative at a point P in the direction of another point Q is computed with $\hat{\mathbf{u}}}=\frac{\overrightarrow{PQ}}{||\overrightarrow{PQ}||}$.

The gradient captures all of the first derivatives of a function in a single vector:


Given a function f(x,y), the gradient of f associates to each point in the domain of f a vector consisting of the partial derivatives of f. The standard notation for the gradient is $\nabla f$, and the formal definition is

\begin{align} \nabla f (x_0,y_0) = \langle f_x(x_0,y_0), f_y(x_0,y_0)\rangle. \end{align}

Multivariable Integration

Iterated Integrals

General Regions

Optimization Problems

The Second Derivative Test

Lagrange Multipliers

More on Multivariable Integration

Other Coordinate Systems

Line Integrals

Surface Integrals

Going Further

Theorems of Vector Calculus

The Road Ahead

real analysis
differential equations


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