Partial Derivatives and Integrals

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Partial Derivatives of Functions from Rn to Rm

One of the core concepts of multivariable calculus involves the various differentiations of functions from to . We begin by defining the concept of a partial derivative of such functions.

Definition: Let be open, , and . Denote for each , i.e., is the unit vector in the direction of the coordinate axis. Then the Partial Derivative of at with Respect to the Variable is defined as provided that this limit exists.

Suppose that is open, , and . Then the partial derivative of at with respect to the variable is:

For example, consider the function defined by:

Then the partial derivative of with respect to the variable at the point is:

We can also easily calculate the partial derivatives and . So the definition of a partial derivative for is somewhat justified since the case when yields the definition of the partial derivative for a multivariable real-valued function.

Furthermore, suppose that and that . Then where for each are single-variable real-valued functions. The partial derivative of with respect to the first variable (the only variable, or simply just the derivative) at is:

For example, consider the function defined by:

Then the derivative of is:

And the derivative of at is:

Once again, the definition is justified since when we have that the definition reduces down to the special case of differentiating a single variable vector-valued function.

Now let's look at a more complicated example of computing a partial derivative. Let be defined by:

Then the partial derivative of at with respect to the first variable is:

So the partial derivative of with respect to the first variable at say is .

The Riemann Integral

Let be a bounded function defined on the closed and bounded interval . Let be a partition of with:

Let denote the set of all partitions on . For each we define:

With the notation above we can define the upper and lower Riemann sums associated with the partition for the function .

Definition: Let be a bounded function on the closed and bounded interval and let . The Upper Riemann Sum Associated with the Partition for is . The Lower Riemann Sum Associated with the Partition for is .

It can be shown that for any partitions with we have that:

So as we consider partitions that are finer and finer, decreases and increases. It can also be shown that for any partitions :

That is, the set of all upper Riemann sums is bounded below by any lower Riemann sum, and the set of all lower Riemann sums is bounded above by any upper Riemann sum. We now define the upper and lower Riemann integrals of a bounded function on .

Definition: Let be a bounded function on the closed and bounded interval . The Upper Riemann Integral of is defined to be
and the Lower Riemann Integral of is defined to be
.

Another way t o define the upper and lower Riemann integrals of is through step functions. If is a step function defined on then it is easy to show that the upper and lower Riemann integrals of exist and define the upper and lower Riemann integrals of to also be:

We are finally able to define what it means for a bounded function defined on a closed and bounded interval to be Riemann integrable.

Definition: Let be a bounded function on the closed and bounded interval . Then is said to be Riemann Integrable on denoted if .

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