Question

Use the procedure in Exercise 57 to construct potential functions for the following fields. $$\mathbf{F}=\langle-y,-x\rangle$$

Short answer

Based on the step-by-step solution, provide the short answer: The potential function for the given vector field \(\mathbf{F}=\langle-y,-x\rangle\) is \(\phi(x,y)=-yx+C\), where C is a constant.

Step-by-step solution

Verify if the Curl of F is zero

In order to check if F has a potential function, we need to compute the curl of F and check if it is zero. If curl(F) = 0, F is conservative and has a potential function. The curl of a vector field with components (P, Q) is given by: $$\text{curl}(F)=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}$$ For our given vector field, \(\mathbf{F}=\langle-y,-x\rangle\), P = -y and Q = -x. So, we can compute the curl as: $$\text{curl}(F)=\frac{\partial (-x)}{\partial x}-\frac{\partial (-y)}{\partial y}$$ $$\text{curl}(F)= -1- (-1) = 0$$ Since curl(F) = 0, F has a potential function.

Integrate P with respect to x to find an intermediate potential function

To find the potential function, let's first integrate P = -y with respect to x: $$\int -y\,dx = -yx + g(y)$$ Here, g(y) is an arbitrary function of y only.

Partially differentiate the result with respect to y and set equal to Q

Next, we partially differentiate the result from step 2 with respect to y: $$\frac{\partial}{\partial y}(-yx+g(y))=-x+\frac{dg(y)}{dy}$$ Now, set this expression equal to Q = -x and solve for g'(y): $$-x+\frac{dg(y)}{dy}=-x$$ $$\frac{dg(y)}{dy}=0$$

Integrate g'(y) to find g(y) and obtain the potential function

Now, we integrate g'(y) to find g(y): $$\int 0\,dy = C$$ where C is a constant.

Substitute g(y) in our intermediate potential function as obtained in step 2

To find the final potential function, we substitute g(y) in the expression obtained in step 2: $$\phi(x,y)=-yx+C$$ The potential function for the given vector field \(\mathbf{F}\) is: $$\phi(x,y)=-yx+C$$

Key concepts

Potential Functions

A potential function, often denoted as \(\phi(x,y)\), helps to summarize a vector field in a single scalar function. This is incredibly useful because if a vector field \(\mathbf{F}\) has a potential function, it means the field is made by the gradient of this function and is known as a conservative vector field.
To find a potential function, you would typically integrate the components of the vector field. For example, in the vector field \(\mathbf{F}=\langle-y,-x\rangle\), the potential function is found by integrating each component with respect to its variable, ensuring that each integration includes an arbitrary function of the other variable.
The procedure for finding potential functions involves:

• Integrating one component of the vector field with respect to one variable.
• Determining the other variable's influence by differentiation and setting it to match the field component.

This helps us isolate the function \(g(y)\) that holds variables constant during each integration, leading us to the overall scalar potential function for the vector field.

Conservative Vector Fields

A conservative vector field is one where the line integral between any two points is independent of the path taken. These fields are associated with potential energy, where the net work done by the field around a closed path is zero.
Key characteristics of conservative vector fields include:

• The curl is zero, i.e., \(\text{curl}(\mathbf{F}) = 0\)
• Existence of a potential function \(\phi(x,y)\) such that \(\mathbf{F} = abla \phi\)

By checking the vector field \(\mathbf{F}=\langle-y,-x\rangle\), we computed its curl to verify its conservativeness. Since the curl came out to be zero, it means the field has a potential function. Understanding this property is crucial in physics and engineering, as it represents energy-conserving systems.

Curl of a Vector Field

The curl of a vector field provides information about the field's rotation. In two dimensions, the calculation simplifies but still serves an essential purpose in vector calculus and fields.
For a vector field \(\mathbf{F} = \langle P(x,y), Q(x,y) \rangle\), the curl is given by \(\text{curl}(\mathbf{F}) = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\). If the curl equals zero, the vector field is conservative, implying the existence of a potential function.
This operation is fundamental as it helps us identify properties of physical fields such as magnetic or velocity fields in fluids. If a field has rotation, its curl is non-zero, indicating some rotational aspects, whereas a zero curl indicates a potential field without such rotations.