(Continued from part 1.) This post contains some typical problems for multi-variable calculus. Throughout them, sufficient regularity is always presumed.

## Topic 4. Functions of Several Variables (Cont.)

For a function of several variables, one can calculate its derivative in a certain direction. In particular, the directional derivatives in coordinate directions are called partial derivatives. All partial derivatives of the function form a vector called **gradient** of the function . Gradient of a (scalar) function of variables is a vector-valued function of variables.

With gradient notation, one can write the **linear approximation** of around (first-order Taylor expansion) as

$$ f(x) = f(\mathbf{x}_0) + \nabla f(\mathbf{x}_0) (\mathbf{x} – \mathbf{x}_0) + O(|\mathbf{x} – \mathbf{x}_0|^2).$$

Note that the linear part of the expansion $$ g(x) = f(\mathbf{x}_0) + \nabla f(\mathbf{x}_0) (\mathbf{x} – \mathbf{x}_0) $$ is a function whose graph being the **tangent plane** of the graph of at .

In addition, when given an **equation** , with , it defines a relation among components of . Whenever , the equation fixes the value of $x_i$ as an implicit function locally around (**inverse function theorem**).

Geometrically, the equation defines a surface in . The tangent plane of such a surface can be also given by manipulating the gradient of the function:

$$ 0 = \nabla f(\mathbf{x}_0) (\mathbf{x} – \mathbf{x}_0). $$

Note that here is a point on the surface, i.e., .

## Topic 5. Maximum/Minimum Problems for Functions of Several Variables

As with functions of single variable, to solve problems of the form

$$ \max _{\mathbf{x} \in D, D \subset \mathbb{R}^n} f(\mathbf{x}) \text{ or } \min _{\mathbf{x} \in D, D \subset \mathbb{R}^n} f(\mathbf{x}), $$

one needs to divide the problem into sub-problems by investigating

- Critical points of which solves .
- Points on .

Unlike functions of a single variable, critical points of a function of several variables may be local maxima, local minima, or saddle point, depending on eigenvalues of the Hessian matrix. When the Hessian matrix is non-degenerate, a critical point is a

- Local minima, if all eigenvalues are non-negative.
- Local maxima, if all eigenvalues are non-positive.
- Saddle point, if eigenvalues are of mixed signs.

## Topic 6. Multiple Integration

Single integration calculates the enclosed area of the graph of the function within interval . Likewise, double integration calculate the enclose volume of the graph of the function within region . Similar concept of “**volume**” can be generalized to arbitrary dimension, yielding analogous multiple integration , where .

To actually calculate the value of multiple integrations, it is usually convenient to convert them to **iterated integrals**, which can be then calculated iteratively using the fundamental theorem of calculus. This process is justified by Fubini’s theorem

## Topic 6. Vector Calculus

Vector fields are functions that map a position in space to a direction ().

A vector field is called a **gradient field** (also called **conservative field**) if there exists a potential function for such that

$$ \mathbf{F} = \nabla \phi. $$

Force fields in physics can be modeled using vector fields. When a particle travels through space along a curve subject to the force field, the work done throughout the process is given by

$$ W = \int_C \mathbf{F}(\mathbf{r}(s)) \cdot d \mathbf{r}(s) = \int_C \mathbf{F}(\mathbf{r}(s)) \cdot \mathbf{r}'(s) ds.$$

This is a scalar integration that can be calculated directly.

Specially, when is conservative,

$$ \mathbf{F}(\mathbf{r}(s)) \cdot d \mathbf{r}(s) = \nabla \phi (\mathbf{r}(s)) \cdot d \mathbf{r}(s) = d \phi (\mathbf{r}(s)). $$

That is, given a conservative force field, the work is equal to

$$ W = \int_C d \phi (\mathbf{r}(s)) = \phi(\mathbf{x}_e) – \phi(\mathbf{x}_s), $$

which only depends on the starting point and the ending point (and independent of the path).