Understanding the Reverse of Partial Differentiation

Partial differentiation is a fundamental concept in multivariable calculus, used when dealing with functions of multiple variables. However, what about the reverse operation?

What is Partial Differentiation?

Partial differentiation involves finding the derivative of a function with respect to one variable while keeping the other variables constant. For instance, consider the function:

z x * y

This function represents a plane with a double slope. Its partial derivatives with respect to x and y are both 1. When we fix y to a constant y0, the partial integral to x is given by:

w° x * y0

This represents a plane with a single slope along x. Similarly, when fixing x to a constant x0, the partial integral with respect to y is:

w°° x0 * y

Both these planes do not coincide with the original plane but intersect it along lines passing through the points (0, y0) and (x0, 0).

Contour Integrals and Partial Integrals

While partial derivatives handle functions with multiple variables, contour or path integrals allow us to integrate over curves in multi-dimensional spaces. These are useful in understanding the reverse operation of partial differentiation.

The Concept of Partial Integration

Purely speaking, the reverse of partial differentiation is partial integration. However, this term can sometimes be confused with integration by parts. Let's explore an example:

Consider the partial derivative: dz/dx 3x^2y. To reverse this, we would integrate with respect to x, treating y as a constant:

d(x^3 * (1/3z - 2/3y)) / dx 3x^2y

The result shows that the partial integration operation gives back the original function, but with a constant of integration, which is often denoted as C.

Partial Integration in Vector Fields

A partial derivative can be thought of as a component of a vector field. The integral curve, defined by the vector field, follows the direction of the vector field. Hence, integrating a partial derivative is akin to integrating a component of a vector field to find the original function.

For instance, if the partial derivative of f is g, then integrating g with respect to x can recover the function f up to a constant. This is reminiscent of the idea of integral curves in vector calculus.

Double and Triple Integrals

While the direct reverse of partial differentiation is partial integration, often, when dealing with multivariable functions, the integration is done over areas or volumes. This is achieved through the use of double and triple integrals.

For a function of two variables u f(x, y), the double integral over a region R would be:

INT_R f(x, y) dA

Similarly, for a function of three variables u f(x, y, z), the triple integral over a volume V would be:

INT_V f(x, y, z) dV

These integrals are used to find the total value of the function over the specified domain.

Conclusion

In summary, the reverse of partial differentiation can be termed as partial integration. This operation is crucial in understanding multivariable calculus and the integration of functions over regions and volumes. While the term partial integration might sometimes be confused with integration by parts, the context usually clarifies the meaning.