Calculus/Derivatives of multivariate functions

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Derivatives of multivariate functions

The matrix of a linear transformationEdit

Theorem

A linear transformation   amounts to multiplication by a uniquely defined matrix; that is, there exists a unique matrix   such that

 
Proof

We set the column vectors

 

where   is the standard basis of   . Then we define from this

 

and note that for any vector   of   we obtain

 

Thus, we have shown existence. To prove uniqueness, suppose there were any other matrix   with the property that   . Then in particular,

 

which already implies that   (since all the columns of both matrices are identical). 

How to generalise the derivativeEdit

It is not immediately straightforward how one would generalize the derivative to higher dimensions. For, if we take the definition of the derivative at a point  

 

and insert vectors for   and   , we would divide the whole thing by a vector. But this is not defined.

Hence, we shall rephrase the definition of the derivative a bit and cast it into a form where it can be generalized to higher dimensions.

Theorem

Let   be a one-dimensional function and let   . Then   is differentiable at   if and only if there exists a linear function   such that

 

We note that according to the above, linear functions   are given by multiplication by a  -matrix, that is, a scalar.

Proof

First assume that   is differentiable at  . We set   and obtain

 

which converges to 0 due to the definition of   .

Assume now that we are given an   such that

 

Let   be the scalar associated to   . Then by an analogous computation   . 

With the latter formulation of differentiability from the above theorem, we may readily generalize to higher dimensions, since division by the Euclidean norm of a vector is defined, and linear mappings are also defined in higher dimensions.

Definition

A function   is called differentiable or totally differentiable at a point   if and only if there exists a linear function   such that

 

We have already proven that this definition coincides with the usual one in the one-dim. case (that is  ).

We have the following theorem:

Theorem

Let   be a set, let   be an interior point of   , and let   be a function differentiable at   . Then the linear map   such that

 

is unique; that is, there exists only one such map   .

Proof

Since   is an interior point of  , we find   such that   . Let now   be any other linear mapping with the property that

 

We note that for all vectors of the standard basis   , the numbers   for   are contained within   . Hence, we obtain by the triangle inequality

 

Taking   , we see that   . Thus,   and   coincide on all basis vectors, and since every other vector can be expressed as a linear combination of those, by linearity of   and   we obtain   . 

Thus, the following definition is justified:

Definition

Let   be a function (where   is a subset of  ), and let   be an interior point of   such that   is differentiable at   . Then the unique linear function   such that

 

is called the differential of   at   and is denoted   .

Directional and partial derivativesEdit

We shall first define directional derivatives.

Definition

Let   be a function, and let   be a vector. If the limit

 

exists, it is called directional derivative of   in direction   . We denote it by   .

The following theorem relates directional derivatives and the differential of a totally differentiable function:

Theorem

Let   be a function that is totally differentiable at  , and let   be a nonzero vector. Then   exists and is equal to   .

Proof

According to the very definition of total differentiability,

 

Hence,

 

by multiplying the above equation by   . Noting that

 

the theorem follows. 

A special case of directional derivatives are partial derivatives:

Definition

Let   be the standard basis of   , let   and let   be a function such that the directional derivatives   all exist. Then we set

 

and call it the partial derivative in the direction of   .

In fact, by writing down the definition of   , we see that the partial derivative in the direction of   is nothing else than the derivative of the function   in the variable   at the place   . That is, for instance, if

 

then

 

that is, when forming a partial derivative, we regard the other variables as constant and derive only with respect to the variable we are considering.

The Jacobian matrixEdit

From the above, we know that the differential of a function   has an associated matrix representing the linear map thus defined. Under a condition, we can determine this matrix from the partial derivatives of the component functions.

Theorem

Let   be a function such that all partial derivatives exist at   and are continuous in each component on   for a possibly very small, but positive   . Then   is totally differentiable at   and the differential of   is given by left multiplication by the matrix

 

where   .

The matrix   is called the Jacobian matrix.

Proof
   
 

We shall now prove that all summands of the last sum go to 0.

Indeed, let   . Writing again   , we obtain by the one-dimensional mean value theorem, first applied in the first variable, then in the second and so on, the succession of equations

 
 
 
 

for suitably chosen   . We can now sum all these equations together to obtain

 

Let now   . Using the continuity of the   on   , we may choose   such that

 

for   , given that   (which we may assume as  ). Hence, we obtain

 

and thus the theorem. 

Corollary

If   is continuously differentiable at   and   , then

 
Proof