
MSC Classification: msc26 (Real functions) 
Prerequisites: Calculus I, Vectors and Space
Getting Oriented
Rough Guides to Analysis 
Calculus  
Real Analysis  
Complex Analysis 
Multivariable calculus extends the notions of singlevariable calculus to functions with more than one input and/or output.
The Basics
Multivariable Functions
Singlevariable 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(x_{1}, x_{2}, …, x_{n}). 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 R^{n+1}, and contour plots exist in R^{n}. 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=y_{0}, 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 y_{0}=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=y_{0}. Analogous definitions hold in the case where x is fixed to x_{0}.
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 xyplane. 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=y_{0} is fixed, then one writes $\frac{\Delta f}{\Delta x}\Big_{y=y_0}$. If just x is changing and x=x_{0} is fixed, then one writes $\frac{\Delta f}{\Delta y}\Big_{x=x_0}$. If both x and y are changing, rateofchange 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:
(1)The definition of multivariable derivatives is essentially the same. The difference lies in the choice of which variable is changing.
Definition
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
(2)Both notations f_{x} 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:
(3)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.
Definition
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
(4)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 nonunit 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:
Definition
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
(5)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