Problem 27 - Calculus Early Transcendentals

Question

Find \(\nabla \times(\nabla \times \mathbf{F})\) $$ \mathbf{F}(x, y, z)=x y \mathbf{j}+x y z \mathbf{k} $$

Step-by-Step Solution

Answer

The result of \(\nabla \times(\nabla \times \mathbf{F})\) is \(x \mathbf{i} + y \mathbf{j} - 2z \mathbf{k}\).

Step 1: Identify the Components of \(\mathbf{F}\)

The vector field \(\mathbf{F}(x, y, z)\) is given as \(\mathbf{F} = x y \mathbf{j} + x y z \mathbf{k}\). This means it has no \(\mathbf{i}\) component, a \(\mathbf{j}\) component of \(xy\), and a \(\mathbf{k}\) component of \(xyz\).

Step 2: Compute \(\nabla \times \mathbf{F}\)

The curl of \(\mathbf{F}\), \(abla \times \mathbf{F}\), is computed using:\[abla \times \mathbf{F} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \0 & xy & xyz\end{vmatrix}\]This determinant is expanded to find:\[abla \times \mathbf{F} = \left( \frac{\partial (xyz)}{\partial y} - \frac{\partial (xy)}{\partial z} \right) \mathbf{i} + \left( \frac{\partial (0)}{\partial z} - \frac{\partial (xyz)}{\partial x} \right) \mathbf{j} + \left( \frac{\partial (xy)}{\partial x} - \frac{\partial (0)}{\partial y} \right) \mathbf{k} \]This simplifies to:\[abla \times \mathbf{F} = yz \mathbf{i} - xz \mathbf{j} + y \mathbf{k} \]

Step 3: Compute \(\nabla \times (\nabla \times \mathbf{F})\)

To find the curl of the curl of \(\mathbf{F}\), compute \(abla \times (abla \times \mathbf{F})\):\[abla \times (abla \times \mathbf{F}) = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \yz & -xz & y\end{vmatrix}\]Expanding this determinant gives:\[abla \times (abla \times \mathbf{F}) = \left( \frac{\partial y}{\partial y} - \frac{\partial (-xz)}{\partial z} \right) \mathbf{i} + \left( \frac{\partial (yz)}{\partial z} - \frac{\partial y}{\partial x} \right) \mathbf{j} + \left( \frac{\partial (-xz)}{\partial x} - \frac{\partial (yz)}{\partial y} \right) \mathbf{k} \]Simplifying, we get:\[abla \times (abla \times \mathbf{F}) = (0 + x) \mathbf{i} + (y - 0) \mathbf{j} + (-z - z) \mathbf{k} \]This simplifies further to:\[abla \times (abla \times \mathbf{F}) = x \mathbf{i} + y \mathbf{j} - 2z \mathbf{k} \]

Step 4: Final Result Interpretation

The final expression \(x \mathbf{i} + y \mathbf{j} - 2z \mathbf{k}\) is the result of the operation \(abla \times (abla \times \mathbf{F})\). This result shows how the initial vector field \(\mathbf{F}\) is altered through this double curl operation.

Key Concepts

Understanding Curl

Curl is an important concept in vector calculus. It helps determine how a vector field twists or rotates in space. Think of it as the field's tendency to swirl around a point.
This makes it especially significant in physics, particularly in fluid dynamics and electromagnetism.

### Calculating the Curl
To find the curl of a vector field, we use the cross product of the del operator \(abla\) with the vector field \(\mathbf{F}\).
If \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\), the curl is computed as:
\[abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix}\]
This determinant method helps extract the rotating components from the original vector field. In the example, the curl of \(\mathbf{F}\) was calculated, leading to a new vector field depicting circular flow.

Exploring Vector Fields

Vector fields are mathematical structures that assign a vector to every point in space. Imagine each point in space having a little arrow indicating direction and magnitude.
These are widely used to represent diverse physical phenomena, such as magnetic fields, fluid flow, or wind patterns.

### Components of Vector Fields
The vector field \(\mathbf{F}(x, y, z) = xy \mathbf{j} + xyz \mathbf{k}\) has:

• No \(\mathbf{i}\) component;
• A \(\mathbf{j}\) component of \(xy\);
• A \(\mathbf{k}\) component of \(xyz\).

This means it points in the \(\mathbf{j}\) and \(\mathbf{k}\) directions, with direction and magnitude changing depending on \(x\), \(y\), and \(z\) values. Understanding these components is crucial to visualizing how a vector field behaves across different regions of space.

The Significance of Gradient

A gradient in vector calculus represents how a scalar field changes in space. It's like a map that shows the steepness of a hill at every point. For a scalar function \(f(x, y, z)\), the gradient is a vector field given by:
\[ abla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} \]
This shows the direction of the steepest ascent of the function, pointing from regions of lower value to higher value.

### Differences from Curl
While curl measures rotational effects, a gradient points towards areas of increase. Gradients are often used in optimization problems, determining how to "climb the hill" most efficiently. This makes them useful in fields like economics, engineering, and more, where understanding how quantities change is crucial.