Problem 19 - Calculus Early Transcendentals

Question

Find div F and curl F. $$ \mathbf{F}(x, y, z)=7 y^{3} z^{2} \mathbf{i}-8 x^{2} z^{5} \mathbf{j}-3 x y^{4} \mathbf{k} $$

Step-by-Step Solution

Answer

\( \text{div} \mathbf{F} = 0 \); \( \nabla \times \mathbf{F} = (40x^2 z^4 - 12xy^3)\mathbf{i} + (14y^3 z + 3y^4)\mathbf{j} + (-16xz^5 - 21y^2 z^2)\mathbf{k} \).

Step 1: Understand the Vector Field

The given vector field is \( \mathbf{F}(x, y, z) = 7 y^3 z^2 \mathbf{i} - 8 x^2 z^5 \mathbf{j} - 3 x y^4 \mathbf{k} \). It consists of components that are functions of \( x \), \( y \), and \( z \). We need to find the divergence and curl of this vector field.

Step 2: Recall the Divergence Formula

The divergence of a vector field \( \mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by the formula \( \text{div} \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).

Step 3: Calculate Partial Derivatives for Divergence

For \( \text{div} \mathbf{F} \):* \( \frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(7 y^3 z^2) = 0 \) because there is no \( x \) term.* \( \frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(-8 x^2 z^5) = 0 \) because there is no \( y \) term.* \( \frac{\partial R}{\partial z} = \frac{\partial}{\partial z}(-3 x y^4) = 0 \) because there is no \( z \) term.

Step 4: Calculate Divergence

Add up the results from Step 3:\[ \text{div} \mathbf{F} = 0 + 0 + 0 = 0 \]

Step 5: Recall the Curl Formula

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 6: Calculate Partial Derivatives for Curl

For \( abla \times \mathbf{F} \):* \( \frac{\partial R}{\partial y} = \frac{\partial}{\partial y}(-3 x y^4) = -12 x y^3 \)* \( \frac{\partial Q}{\partial z} = \frac{\partial}{\partial z}(-8 x^2 z^5) = -40 x^2 z^4 \)* \( \frac{\partial P}{\partial z} = \frac{\partial}{\partial z}(7 y^3 z^2) = 14 y^3 z \)* \( \frac{\partial R}{\partial x} = \frac{\partial}{\partial x}(-3 x y^4) = -3 y^4 \)* \( \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(-8 x^2 z^5) = -16 x z^5 \)* \( \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(7 y^3 z^2) = 21 y^2 z^2 \)

Step 7: Calculate Curl

Substitute the calculated partial derivatives from Step 6 into the curl formula:\[abla \times \mathbf{F} = \\left( -12 x y^3 + 40 x^2 z^4 \right) \mathbf{i} + \\left( 14 y^3 z + 3 y^4 \right) \mathbf{j} + \\left( -16 x z^5 - 21 y^2 z^2 \right) \mathbf{k} \]

Step 8: Finalize the Results

Thus, the divergence of the vector field is \( \text{div} \mathbf{F} = 0 \) and the curl of the vector field is \[abla \times \mathbf{F} = \\left( 40 x^2 z^4 - 12 x y^3 \right) \mathbf{i} + \\left( 14 y^3 z + 3 y^4 \right) \mathbf{j} + \\left( -16 x z^5 - 21 y^2 z^2 \right) \mathbf{k} \]

Key Concepts

Divergence

Divergence is a key concept in vector calculus, and it helps us to measure how much a vector field spreads out at a given point. If you imagine a vector field as a grid of arrows, the divergence gives an indication of what happens to the space around a point. Do the arrows point away, converge, or neither? Mathematically, the divergence of a vector field \( abla \cdot \mathbf{F} \) is calculated using partial derivatives.

• The divergence formula for a vector field \( \mathbf{F}(x, y, z)=P \mathbf{i}+Q \mathbf{j}+R \mathbf{k} \) is \( \text{div} \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).
• This means we take the derivative of each component of the vector field with respect to its variable and sum them up.
• If the result is zero, like in our exercise example, the field is said to be solenoidal, indicating that it neither diverges nor converges at any point.
• In practical terms, imagine a fluid flow: No fluid is "escaping" at any point in the field if divergence is zero. The fluid is perfectly balanced inside the space.

Curl

Curl deals with the rotation or the swirling strength of a vector field. If you visualize a vector field on a plane, the curl shows whether the field causes objects to rotate around a point. The mathematical expression for curl uses cross products and partial derivatives.

• The formula to find the curl of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \).
• Each term in this formula involves the difference between two partial derivatives, representing the twisting or turning effect in the field.
• If the curl is zero, the vector field is conservative, which means no rotation at any point.
• In our given vector field, we calculated the curl to be \([ 40x^2z^4 - 12xy^3 ] \mathbf{i} + [ 14y^3z + 3y^4 ] \mathbf{j} + [ -16xz^5 - 21y^2z^2 ] \mathbf{k}\), showing the rotational properties at different points.

Vector Field

A vector field is a concept where each point in a space is associated with a vector. This is like a map that assigns a vector to each coordinate in a space.

• In three-dimensional space, a vector field \( \mathbf{F}(x, y, z) \) has components \( P, Q, \text{and } R \) corresponding to the \textbf{i}, \textbf{j}, and \textbf{k} unit vectors, respectively.
• Vector fields can represent many physical phenomena. They are used to describe forces like magnetic and gravitational fields, fluid flows, and much more.
• The given vector field, \( \mathbf{F}(x, y, z) = 7y^3z^2 \mathbf{i} - 8x^2z^5 \mathbf{j} - 3xy^4 \mathbf{k} \), assigns a vector to every point \( (x, y, z) \).
• Understanding how divergence and curl apply to vector fields helps us analyze the behavior of systems modeled by these fields, seeing how they spread and rotate.