Problem 2 - Thomas Calculus

Question

Which fields are conservative, and which are not? \(\mathbf{F}=(y \sin z) \mathbf{i}+(x \sin z) \mathbf{j}+(x y \cos z) \mathbf{k}\)

Step-by-Step Solution

Answer

The field is conservative as its curl is zero.

Step 1: Understand the Concept

A vector field is conservative if there exists a scalar potential function, \(\), such that \(abla  = extbf{F}\). This means the curl of a conservative vector field is zero. We will verify if the field \(\mathbf{F} = (y \sin z) \mathbf{i} + (x \sin z) \mathbf{j} + (xy \cos z) \mathbf{k}\) is conservative by calculating its curl.

Step 2: Write the Formula for the Curl

The formula for the curl of a vector field \(\mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\) is given by \(abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) \mathbf{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right) \mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \mathbf{k}\).

Step 3: Calculate Partial Derivatives

For \(\mathbf{F} = (y \sin z) \mathbf{i} + (x \sin z) \mathbf{j} + (xy \cos z) \mathbf{k}\):- \(\frac{\partial R}{\partial y} = \frac{\partial}{\partial y}(xy \cos z) = x \cos z\)- \(\frac{\partial Q}{\partial z} = \frac{\partial}{\partial z}(x \sin z) = x \cos z\)- \(\frac{\partial P}{\partial z} = \frac{\partial}{\partial z}(y \sin z) = y \cos z\)- \(\frac{\partial R}{\partial x} = \frac{\partial}{\partial x}(xy \cos z) = y \cos z\)- \(\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(x \sin z) = \sin z\)- \(\frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(y \sin z) = \sin z\)

Step 4: Compute the Curl

Now plug in the partial derivatives into the curl formula:\(abla \times \mathbf{F} = \left(x \cos z - x \cos z\right) \mathbf{i} + \left(y \cos z - y \cos z\right) \mathbf{j} + \left(\sin z - \sin z\right) \mathbf{k}\)This simplifies to \(abla \times \mathbf{F} = 0 \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k}\).

Key Concepts

Curl of a Vector Field

The curl of a vector field is a concept that helps us determine whether a vector field is conservative. If you imagine a vector field having arrows, representing directions and magnitudes, the curl describes the tendency of these arrows to rotate around a point. It's like observing a whirlpool and measuring how the water spins.

To calculate the curl of a vector field \(\mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\), use the formula for curl given by:

• \(abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) \mathbf{i}\)
• \( + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right) \mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \mathbf{k} \)

When the result of this calculation is zero for all components, it indicates no rotation, suggesting the field is conservative. In other words, if \(abla \times \mathbf{F} = 0\), the field doesn't twist or spin on itself.

Scalar Potential Function

When dealing with conservative vector fields, one of the key concepts is the scalar potential function, often called simply the potential function. This function is like a hidden layer underlying the vector field. It represents a kind of 'gravity' or 'energy' landscape from which the vector field derives direction and magnitude.

If you can find a scalar potential function \(\phi\) such that its gradient equals the vector field \(\mathbf{F} \), then \(\mathbf{F}\) is conservative. Mathematically, this is described by:

• \(abla \phi = \mathbf{F}\)

In simple terms, it's like saying you can retrace all the steps in the vector field back to a single landscape or topology, often related to something like height in physics. Conversely, if no such potential function exists, the vector field is not conservative.

Partial Derivatives

Partial derivatives play a crucial role in not only understanding vector fields but in many areas of calculus and physics. They measure how a function changes as only one of the input variables is modified, keeping the others constant. Think of partial derivatives as looking at the slope, or the rate of change, of a function along one axis at a time.

For a vector field \(\mathbf{F}\), when computing the curl or checking for a scalar potential function, we often need to calculate several partial derivatives. For example, in the vector field \(\mathbf{F} = (y \sin z) \mathbf{i} + (x \sin z) \mathbf{j} + (xy \cos z) \mathbf{k}\), we calculate partial derivatives like \(\frac{\partial R}{\partial y}\) or \(\frac{\partial Q}{\partial x}\). Each of these derivatives helps us understand a specific detail about how the vector field behaves along a particular direction or dimension.

• It's important to be careful with signs and terms since they affect the overall computation of curl.
• Partial derivatives are central in the formulas that determine if a vector field is conservative.

Understanding how to compute and use partial derivatives is essential for mastering vector calculus.