Introduction to Multi-variable Calculus (2)

(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 f(\mathbf{x}) 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 \nabla f(\mathbf{x}). Gradient of a (scalar) function of n variables is a vector-valued function of n variables.

With gradient notation, one can write the linear approximation of f around \mathbf{x}_0 (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 f at \mathbf{x}_0.

In addition, when given an equation f(\mathbf{x})=0, with \mathbf{x} \in \mathbb{R}^n,n>1, it defines a relation among components of \mathbf{x}. Whenever \frac{\partial f}{\partial x_i}(\mathbf{x}_0) \neq 0, the equation fixes the value of $x_i$ as an implicit function locally around \mathbf{x}_0 (inverse function theorem).

Geometrically, the equation f(\mathbf{x})=0 defines a surface in \mathbb{R}^n,n>1. 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 \mathbf{x}_0 is a point on the surface, i.e., f(\mathbf{x}_0)=0.

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 f which solves f(\mathbf{x}) = 0 .
  • Points on \partial D.

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 \int _a ^ b f(x) dx calculates the enclosed area of the graph of the function f within interval [a,b]. Likewise, double integration \int _D f(x,y) dV calculate the enclose volume of the graph of the function f(x,y) within region D. Similar concept of “volume” can be generalized to arbitrary dimension, yielding analogous multiple integration \int _D f(\mathbf{x}) dV , where \mathbf{x}\in \mathbb{R}^n.

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 (\mathbb{R}^n \longrightarrow \mathbb{R}^n).

A vector field \mathbf{F} is called a gradient field (also called conservative field) if there exists a potential function \phi for \mathbf{F} 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 C: \mathbf{x} = \mathbf{r}(s) 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 \mathbf{F} 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 \mathbf{x}_s and the ending point \mathbf{x}_e (and independent of the path).

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