Problem 9 - Calculus: Early Transcendental Functions

Question

Determine whether \(F\) is conservative. If it is, find a potential function \(f.\) $$\mathbf{F}(x, y, z)=\left\langle z^{2}+2 x y, x^{2}+1,2 x z-3\right\rangle$$

Step-by-Step Solution

Answer

The given vector field \(\mathbf{F}(x, y, z)=\left\langle z^{2}+2 x y, x^{2}+1,2 x z-3\right\rangle\) is not conservative.

Step 1: Compute the Curl of F

Firstly, calculate the curl of vector field \(\mathbf{F}.\) The curl of a vector field \(F = \langle P, Q, R \rangle\) in three dimensions is given by the determinant of this matrix: \[ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \P & Q & R\ \end{vmatrix} = \left\langle \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right\rangle \] Hence, \(\nabla \times \mathbf{F} = \langle 0 - 0, 2x - 2x, 2y - 0 \rangle = \langle 0, 0, 2y \rangle\).

Step 2: Check for Conservativeness

From the curl calculated, it can be observed it is not zero. Thus, according to the curl criteria, \(\mathbf{F}\) is not conservative.

Step 3: Statement of Result

The vector field F is not conservative therefore it doesn't have a potential function. Usually, the next step would be to find a potential function \(f\), but in this case, it's not suitable. Therefore, this step is omitted in this case.

Key Concepts

Conservative Vector Fields

A conservative vector field is a special type of vector field where the work done along any closed path is zero. In simpler language, this means that if you start and end at the same point while moving in the field, the total work done is zero. Another key property of conservative vector fields is that they can be expressed as the gradient of a scalar function, known as a potential function.
This implies that if a vector field is conservative, there exists some scalar function \(f\) such that \(\mathbf{F} = abla f\).

• Path Independence: In a conservative vector field, the work done between two points does not depend on the path taken.
• Potential Functions: Conservative vector fields can be represented as gradients of scalar fields.
• irrotational: If the vector field is conservative, the curl of the vector field is zero.

In our exercise, the vector field \(\mathbf{F}(x, y, z)\) was found to have a non-zero curl, indicating that it is not conservative. Thus, it does not satisfy these key properties and lacks a potential function.

Curl of a Vector Field

The curl of a vector field provides a measure of the rotation at every point in the vector field. More specifically, it shows the tendency of the field to circulate around a point. Calculating the curl is an important step in determining whether a vector field is conservative.
The mathematical expression for curl in three dimensions is given as:\[ abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \]This involves calculating the derivatives of the vector field components. If the curl equals zero, the vector field may be conservative.

• Zero Curl: A necessary (but not always sufficient) condition for a vector field to be conservative is that its curl is zero.
• Rotation Visualization: Physically, a non-zero curl implies the presence of some rotation or circulation around a point.

In the problem, the curl of \(\mathbf{F}(x, y, z)\) resulted in \(\langle 0, 0, 2y \rangle\), which is not zero, confirming the vector field is not conservative.

Potential Functions

A potential function is a scalar function whose gradient gives the corresponding vector field. In the context of a conservative vector field, a potential function \(f\) satisfies \(abla f = \mathbf{F}\). Finding a potential function simplifies the calculations of work or energy within the field, as every point in the field can be described by the same scalar function.

• Existence: A potential function exists only if the vector field is conservative.
• Calculation: Finding \(f\) involves integrating the vector field components with respect to their variables.
• Applications: Potential functions are widely used in fields like physics, where forces such as gravity and electromagnetism are naturally conservative.

In this exercise, because the vector field \(\mathbf{F}\) was not conservative, we concluded it was impossible to find a potential function for this particular field.