Problem 24 - Calculus Early Transcendentals

Question

Find \(\nabla \cdot(\mathbf{F} \times \mathbf{G})\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k}} \\ {\mathbf{G}(x, y, z)=x y \mathbf{j}+x y z \mathbf{k}}\end{array} $$

Step-by-Step Solution

Answer

\(-xy^2 - y^2z^2 + x^2y\).

Step 1: Understand the Problem

We need to find the divergence of the cross product of two vectors \( \mathbf{F} \) and \( \mathbf{G} \). Essentially, the task is to compute \( abla \cdot (\mathbf{F} \times \mathbf{G}) \).

Step 2: Recall Useful Vector Identities

One useful vector identity for this problem is the triple product rule for divergence: \( abla \cdot (\mathbf{F} \times \mathbf{G}) = \mathbf{G} \cdot (abla \times \mathbf{F}) - \mathbf{F} \cdot (abla \times \mathbf{G}) \). We will use this identity to simplify the calculation.

Step 3: Compute the Curl of \( \mathbf{F} \)

The curl of a vector \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by \( abla \times \mathbf{F} = (R_y - Q_z)\mathbf{i} + (P_z - R_x)\mathbf{j} + (Q_x - P_y)\mathbf{k} \). For \( \mathbf{F} = yz\mathbf{i} + xz\mathbf{j} + xy\mathbf{k} \), this becomes:\[abla \times \mathbf{F} = (x - z)\mathbf{i} + (-y - 0)\mathbf{j} + (z - z)\mathbf{k} = (x-z)\mathbf{i} - y\mathbf{j}\]

Step 4: Compute the Curl of \( \mathbf{G} \)

Using the same formula for curl, the vector \( \mathbf{G} = 0\mathbf{i} + xy\mathbf{j} + xyz\mathbf{k} \) gives:\[abla \times \mathbf{G} = (xyz_x - (xy)_z)\mathbf{i} + (0 - 0)\mathbf{j} + (0 - x)_z\mathbf{k} = (yz)\mathbf{i} + 0\mathbf{j} - x\mathbf{k}\]

Step 5: Compute \( \mathbf{G} \cdot (\nabla \times \mathbf{F}) \)

This dot product involves vectors \( \mathbf{G} = 0\mathbf{i} + xy\mathbf{j} + xyz\mathbf{k} \) and \( abla \times \mathbf{F} = (x-z)\mathbf{i} - y\mathbf{j} \). Hence,\[\mathbf{G} \cdot (abla \times \mathbf{F}) = 0(x-z) + xy(-y) + xyz(0) = -xy^2\]

Step 6: Compute \( \mathbf{F} \cdot (\nabla \times \mathbf{G}) \)

This dot product involves vectors \( \mathbf{F} = yz\mathbf{i} + xz\mathbf{j} + xy\mathbf{k} \) and \( abla \times \mathbf{G} = yz\mathbf{i} - x\mathbf{k} \). Therefore,\[\mathbf{F} \cdot (abla \times \mathbf{G}) = yz(yz) + xz(0) + xy(-x) = y^2z^2 - x^2y\]

Step 7: Apply the Triple Product Rule

Substitute the calculated dot products into the formula from Step 2: \[abla \cdot (\mathbf{F} \times \mathbf{G}) = \mathbf{G} \cdot (abla \times \mathbf{F}) - \mathbf{F} \cdot (abla \times \mathbf{G})\]\[= (-xy^2) - (y^2z^2 - x^2y)\]\[= -xy^2 - y^2z^2 + x^2y\]

Step 8: Conclusion: Present the Final Result

The divergence of the cross product of the vectors \( \mathbf{F} \) and \( \mathbf{G} \), \( abla \cdot (\mathbf{F} \times \mathbf{G}) \), simplifies to:\[-n\cdot(\mathbf{F} \times \mathbf{G}) = -xy^2 - y^2z^2 + x^2y\]

Key Concepts

Cross Product

A cross product between two vectors is a way to find a vector that is perpendicular to both of them. If you have two vectors, say \( \mathbf{A} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{B} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \), the cross product, \( \mathbf{A} \times \mathbf{B} \), can be determined using the determinant:

• \( \mathbf{i} \) component: \( a_2b_3 - a_3b_2 \)
• \( \mathbf{j} \) component: \( a_3b_1 - a_1b_3 \)
• \( \mathbf{k} \) component: \( a_1b_2 - a_2b_1 \)

The result is another vector perpendicular to both \( \mathbf{A} \) and \( \mathbf{B} \). This operation is super useful in physics for finding things like torques or rotational axes in 3D space. By understanding and effectively calculating cross products, you'll be able to solve a variety of real-world problems involving vectors.

Curl of a Vector

The curl of a vector field measures how much and in which direction the field "twists" at a given point. The formula for the curl of a vector \( \mathbf{F} \) with components \( P, Q, R \) is:\[abla \times \mathbf{F} = (R_y - Q_z)\mathbf{i} + (P_z - R_x)\mathbf{j} + (Q_x - P_y)\mathbf{k}\]This might look tricky at first, but it follows a consistent pattern:

• Imagine coiling ropes or swirling water; the curl describes that type of motion mathematically.
• Geometrically, curl resembles how much the vector field spins around a given point.

Practically, when you compute curl for the vector functions given in our exercise, it simplifies your calculations using vector identities in further steps.

Triple Product Rule

The triple product rule is crucial for simplifying complex vector calculus problems, especially those involving the divergence of cross products. This rule helps transform vector equations into more manageable forms. It states that:\[abla \cdot (\mathbf{F} \times \mathbf{G}) = \mathbf{G} \cdot (abla \times \mathbf{F}) - \mathbf{F} \cdot (abla \times \mathbf{G})\]Here's what makes it handy:

• Breaks down a difficult operation into two simpler dot products.
• Helps link the concepts of divergence, curl, and cross product seamlessly.
• Reduces computational effort by transforming into operations that are often easier to handle.

Understanding and applying this rule allows for more intuitive manipulations of vector fields, aiding in solving engineering and physics problems efficiently.

Vector Identities

Vector identities are mathematical equations that relate different vector operations like dot products, cross products, and curls in useful ways. They simplify calculations and reduce the potential for error in complex algebraic manipulations.Some key vector identities include:

• Distributive properties: \( \mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C} \)
• Cross product identities, such as \( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (\mathbf{A} \cdot \mathbf{C})\mathbf{B} - (\mathbf{A} \cdot \mathbf{B})\mathbf{C} \)
• Triple product rule: relates divergence, curl, and cross products easily, as seen in the problem solution.

These identities are essential tools for vector calculus, and mastering them can make solving vector equations faster and less prone to mistakes. They provide shortcuts and alternative perspectives for understanding complex vector problems.