Mean-Value Theorems for Vector Valued Functions

From Department of Mathematics at UTSA
Jump to navigation Jump to search

Recall that if is a continuous function on the closed interval and differentiable on the open interval (where we assume ) then there exists a number for which:

We would like to generalize this extremely important result to differentiable functions from to . Doing so is actually not that straightforward though. The equation above does not immediately generalize to differentiable functions from to and we will need to do some more work in order to make a meaningful generalization.

To emphasize this, consider the function defined for all by:

Then the total derivative of at evaluated at any is:

Therefore we have that:

And also:

Now set and . Then will always equal the zero vector, , and will never equal the zero vector for any choice of between and . Therefore we see that in general.

Theorem 1 (The Mean Value Theorem): Let be open and let be differentiable on all of . Let be such that the line segment connecting these two points is contained in , i.e., . Then for every there exists a point such that .

In the following Theorem we use the notation "" to denote the line segment that joints the point to . This line segment can be parameterized as .

  • Proof: Let and define a new function for all by:
  • Since is differentiable on we have from the Differentiable Functions from Rn to Rm are Continuous page that is continuous on and so must continuous on . Furthermore, is differentiable on by the chain rule:
  • So by the Mean Value Theorem for single-variable real-valued functions, for and there exists a number for which:
  • The lefthand side of is:
  • The righthand side of is:
  • Set . Then and we have from the equality at that:

Licensing

Content obtained and/or adapted from: