Problem 32 - Applied Calculus

Question

Is there a function \(f\) which has the following partial derivatives? If so what is it? Are there any others? $$ \begin{array}{l} f_{x}(x, y)=4 x^{3} y^{2}-3 y^{4} \\ f_{y}(x, y)=2 x^{4} y-12 x y^{3} . \end{array} $$

Step-by-Step Solution

Answer

Yes, the function is \( f(x, y) = x^4y^2 - 3xy^4 + C \), with \( C \) being any constant.

Step 1: Verify condition for potential function

In order for a function \( f(x, y) \) to exist with the given partial derivatives, the mixed partial derivative conditions must be satisfied. This condition states that \( \frac{\partial}{\partial y} f_x = \frac{\partial}{\partial x} f_y \). Calculate both sides:- Find \( \frac{\partial}{\partial y} (4x^3y^2 - 3y^4) = 8x^3y - 12y^3 \).- Find \( \frac{\partial}{\partial x} (2x^4y - 12xy^3) = 8x^3y - 12y^3 \).Since both are equal, a potential function \( f(x, y) \) can exist.

Step 2: Integrate with respect to x

To find the potential function \( f(x, y) \), integrate \( f_x(x, y) = 4x^3y^2 - 3y^4 \) with respect to \( x \):\[ f(x, y) = \int (4x^3y^2 - 3y^4) \, dx = x^4y^2 - 3xy^4 + g(y) \]Here, \( g(y) \) is an arbitrary function of \( y \) that acts as a constant during partial differentiation with respect to \( x \).

Step 3: Integrate with respect to y

Next, find \( g(y) \) by differentiating the partial potential function with respect to \( y \) and comparing it to \( f_y \):- Differentiate \( f(x, y) = x^4y^2 - 3xy^4 + g(y) \): \[ \frac{\partial}{\partial y}(x^4y^2 - 3xy^4 + g(y)) = 2x^4y - 12xy^3 + g'(y) \]- Set this equal to \( f_y(x, y) = 2x^4y - 12xy^3 \): \[ 2x^4y - 12xy^3 + g'(y) = 2x^4y - 12xy^3 \] \[ g'(y) = 0 \]Thus, \( g(y) = C \), where \( C \) is a constant.

Step 4: Write the potential function f(x, y)

Having determined \( g(y) = C \), the potential function is:\[ f(x, y) = x^4y^2 - 3xy^4 + C \]Thus, \( f(x, y) \) is a function which has the given partial derivatives.

Step 5: Determine other possible functions

The function \( f(x, y) = x^4y^2 - 3xy^4 + C \) includes an arbitrary constant \( C \), which means there are infinitely many potential functions differing only by this constant.

Key Concepts

Potential Function

A potential function is a fundamental concept in mathematics, especially in vector calculus and differential equations. In simple terms, a potential function, often denoted as \( f(x, y) \), is a scalar field whose gradient gives a vector field. In the context of partial derivatives:

• The partial derivative \( f_x(x, y) \) is the rate of change of the function \( f \) along the direction of the \( x \)-axis.
• The partial derivative \( f_y(x, y) \) is the rate of change of the function \( f \) along the direction of the \( y \)-axis.

To determine whether a potential function exists for given partial derivatives, we must ensure these derivatives satisfy a consistency condition, known as the mixed partial derivatives condition (further explained below). If potential function conditions are met, the two partial derivatives can be considered as representations of the slopes of this potential function in their respective directions.

Integration

Integration is the process of finding the whole from its parts, the opposite of differentiation. In the case of partial derivatives, integration allows us to reconstruct the original potential function from its derivatives. In the problem we're examining, we're tasked with finding a potential function \( f(x, y) \) from the given partial derivatives \( f_x \) and \( f_y \).
To find \( f(x, y) \), follow these steps:

• Integrate \( f_x(x, y) \) with respect to \( x \), treating \( y \) as a constant. The integration will yield a function that includes an unknown function \( g(y) \), capturing parts of the potential function that depend only on \( y \).
• Next, to determine \( g(y) \), differentiate the resulting function with respect to \( y \) and set the expression equal to \( f_y(x, y) \). Solving for \( g(y) \) will often give a constant, which is arbitrary and thus denoted as \( C \).

This integration process is crucial because it allows us to ascertain the form of the potential function and understand how all the terms relate to each individual variable.

Mixed Partial Derivatives Condition

The Mixed Partial Derivatives Condition is a criterion that helps ensure a smooth transition from partial derivatives back to a complete function. This condition states that, under usual conditions, the mixed partial derivatives of a function with continuous second derivatives should be equal:

• Formally, this means \( \frac{\partial}{\partial y} f_x(x, y) = \frac{\partial}{\partial x} f_y(x, y) \).

This condition is central when determining existence and uniqueness of potential functions based on given partial derivatives. In simpler terms, it ensures that the order of taking partial derivatives does not matter, confirming the function \( f \) is well-behaved and that a matching potential exists.In practice, for the function \( f(x, y) \) to exist, the verification of this condition involves finding the derivative of \( f_x \) with respect to \( y \) and checking if it equals the derivative of \( f_y \) with respect to \( x \). If both are equal, potential functions can exist, and this was confirmed for our exercise, setting the ground to proceed with integration.