Multivariable Calculus

Multivariable calculus is the study of problems and solutions of continuous functions of more than a single variable. It extends to Vector Analysis and has applications in a wide variety of fields, most notably physics, but also extends to include statistics and finance, biology, and a many other subjects.

Partial DerivativesEdit

If f is a function of more than a single variable we can allow one variable to vary and hold the rest stationary. Differentiating with respect to the one free variable we obtain a partial derivative.

ExamplesEdit

Given a function
 
 
This is the partial derivative of f with respect to x. In each term we hold any variable other than x constant, and differentiate with respect with x.
 
 

Geometry of Partial DerivativesEdit

if f(x,y) is a surface in  , then   is the slope of the tangent line (or rate of change_ of the curve traced by the intersection of the plane y=b, and the surface f, in the direction parallel to the x-axis, at the point (a,b).

the function   represents the rate of change of the family of curves traced by the intersection of the planes generated over the domain of y, and the surface f, at the point x.

SymmetryEdit

It is important to notice and provided without proof, that mixed partial second derivatives of a function are symmetric, given the function has continuous second partial derivatives on the disk that contains the region the function is differentiated over. For a function of 2 variable this means:
 

From this further symmetry relations of higher order functions and their derivatives can be derived. For example, f is a function of 3 variables:

fijk = fikj = fjik = fjki

Extrema of FunctionsEdit

Given a point   that satisfies

 

  is called a critical point of  .
We define the quantity   as
 
If   and  , there is a local minimum at  .
If   and  , there is a local maximum at  .
If  ,   is a saddle point.
If  , further analysis is needed to determine if   is a local minimum, a local maximum, or a saddle point.

Multiple IntegralsEdit

Vector AnalysisEdit

Motivating ExamplesEdit

OptimizationEdit

We study the following function of   and   to determine its critical points and their nature:

 

To locate the critical points of  , we begin by calculating its first partial derivatives and setting them equal to  :

(1)  
(2)  

This is a nonlinear system of equations, but it's fairly easy to solve, as solving for   in (1) and plugging into (2) yields the following quadratic equation in  :

(3)  

The solutions of (3) are   and  , and since we know that   from (1), the two critical points of the function turn out to be   and  . It is now time to determine the nature of these two critical points. We begin by calculating  :

 
 
 
 

For our first critical point  , we have:

 
 

and thus we conclude that there is a local minimum at  .

For the point  ,

 
 

and since now  , we conclude that   is a saddle point of  .

We now analyze an economics example that uses the method of Lagrange multipliers to optimize a given cost function subject to a production constraint.

In particular, let   be our cost function subject to the production constraint  , where   is the number of units being manufactured, and we wish to know what values of   (amount of physical capital) and   (quantity of labor used) minimize the cost.

We set up the Lagrange function   with Lagrange multiplier   as follows:

 

and now we set the gradient of   equal to zero to find its critical points.

 

 

 

From the third equation, we get the two (equivalent) equations

 

and now we plug these two equations into our first two equations for   and   to yield two equations with two unknowns,   and  . Upon solving them (it is not hard), we obtain

 

and

 

The minimum cost is thus