Problem 17 - Calculus: Early Transcendental Functions

Question

Determine whether the given vector field is conservative and/or incompressible. $$\left(\sin z, z^{2} e^{y z^{2}}, x \cos z+2 y z e^{y z^{2}}\right)$$

Step-by-Step Solution

Answer

Perform the above steps to determine whether the given vector field is conservative and/or incompressible. The concrete calculations and their results will reveal whether the vector field is conservative (if its curl vanishes) and incompressible (if its divergence equals zero). Note that this is a general procedure and the specifics and results will depend on the specific form of the vector field.

Step 1: Calculate Curl

The curl of a vector field \(F(x,y,z) =\begin{bmatrix}F_1(x,y,z) \F_2(x,y,z) \F_3(x,y,z) \end{bmatrix}\), is given by \( \nabla \times F = \det(\begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_1(x, y, z) & F_2(x, y, z) & F_3(x, y, z) \ \end{bmatrix}) \). For the given vector field \( F = \begin{bmatrix} \sin z \ z^{2} e^{y z^{2}} \ x \cos z+2 y z e^{y z^{2}} \end{bmatrix} \), calculate \(\nabla \times F\).

Step 2: Check if Curl Vanishes

If the curl of a vector field vanishes (i.e., equals zero), the vector field is conservative. Examine the result of your calculation from step 1.

Step 3: Calculate Divergence

The divergence of a vector field \(F(x,y,z)\) is given by \( \nabla \cdot F = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \). For the given vector field \(F = \begin{bmatrix} \sin z \ z^{2} e^{y z^{2}} \ x \cos z+2 y z e^{y z^{2}} \end{bmatrix}\), calculate \( \nabla \cdot F\).

Step 4: Check if Divergence Equals Zero

If the divergence of a vector field equals zero, the vector field is incompressible. Examine the result of your calculation from step 3.

Key Concepts

Conservative Vector Field

A vector field is considered conservative if it represents the gradient of some scalar function, commonly referred to as the potential function. In simpler terms, it means that if you were to move a particle through the field along a closed path, the net work done would be zero. This property makes conservative fields particularly interesting in physics, especially in the study of potential energy.

To determine if a vector field is conservative, you can calculate the curl of the vector field. If it is zero everywhere in the domain, the field is conservative. With the example vector field \( F(x, y, z) = (\sin(z), z^2 e^{yz^2}, x \cos(z) + 2yz e^{yz^2}) \) from the exercise, the curl was computed. If this curl calculation yields a zero vector everywhere, then the original vector field is indeed conservative. It's essential that the domain of the vector field is simply connected; otherwise, the curl being zero isn't a sufficient condition for the field to be conservative.

Incompressible Vector Field

In contrast to conservative fields, incompressible vector fields are all about volume preservation. They are analogous to fluids that don't change in density; no matter how the fluid moves or flows, the same amount of fluid occupies the same amount of space.

The mathematical condition for incompressibility is tied to the divergence of the vector field. To check if a given vector field is incompressible, you calculate its divergence. When the divergence equals zero at all points, the vector field is incompressible. In the case of the vector field from the exercise, obtaining a zero divergence after computation indicates that the field is incompressible, meaning that it has a constant density throughout its domain.

Curl of a Vector Field

The curl is a measure of the rotation or swirl of a vector field around a point. Imagine a paddle wheel placed in a fluid flow; the way it would spin tells you the curl at that point. Calculating the curl involves partial derivatives and the cross product, leading to a new vector that represents the rotation intensity and direction.

To calculate it, apply the determinant to a special matrix consisting of unit vectors, partial derivative operators, and the components of the field itself, as shown in the example solution. If the resulting vector from this calculation is non-zero, it means the vector field has some rotation around that point. If the curl is zero, it suggests there's no rotation, and the vector field might be conservative (if additional conditions are met).

Divergence of a Vector Field

Divergence is a scalar representing the magnitude of a vector field's source or sink at a given point, very much like how a balloon inflates or deflates. If you're dealing with fluid flow, for example, a positive divergence indicates fluid is emanating from a point, whereas a negative divergence implies fluid is being drawn into a point.

Computing the divergence involves summing the partial derivatives of the field's components with respect to their own variables. With the exercise vector field \( F(x, y, z) \) in mind, finding the divergence gives you an insight into the field's compressibility; as explained earlier, a divergence of zero across the field confirms it is incompressible. This concept is foundational in fluid dynamics and electromagnetism.