Problem 16 - Advanced Engineering Mathematics

Question

In Problems \(7-16\), find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=x^{2} \sin y z \mathbf{i}+z \cos x z^{3} \mathbf{j}+y e^{5 x y} \mathbf{k} $$

Step-by-Step Solution

Answer

The curl is \((-3xz^3 \sin(xz^3) + 5xy e^{5xy}) \mathbf{i} - (5y^2 e^{5xy} - x^2 y \cos(yz)) \mathbf{j} - (z^4 \sin(xz^3) + x^2 z \cos(yz)) \mathbf{k}\), and the divergence is \(2x \sin(yz)\).

Step 1: Understanding the Vector Field

The given vector field is a vector function \( \mathbf{F}(x, y, z) = x^2 \sin(yz) \mathbf{i} + z \cos(xz^3) \mathbf{j} + y e^{5xy} \mathbf{k} \). This field has three components: \(F_1 = x^2 \sin(yz)\), \(F_2 = z \cos(xz^3)\), and \(F_3 = y e^{5xy}\). We are required to find both the curl and divergence of this vector field.

Step 2: Calculating the Curl

The curl of a vector field \( \mathbf{F} = (F_1, F_2, F_3) \) is given by the determinant of the matrix formed with the unit vectors \( (\mathbf{i}, \mathbf{j}, \mathbf{k}) \), partial derivative operators \( (\partial / \partial x, \partial / \partial y, \partial / \partial z) \), and \( (F_1, F_2, F_3) \):\[abla \times \mathbf{F} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \F_1 & F_2 & F_3 \\end{vmatrix}\]Calculate this determinant to find the curl: \[abla \times \mathbf{F} = \left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}\right) \mathbf{i} - \left(\frac{\partial F_3}{\partial x} - \frac{\partial F_1}{\partial z}\right) \mathbf{j} +\left(\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right) \mathbf{k}\].

Step 3: Simplifying Curl Components

To find individual components of the curl:1. \(\frac{\partial F_3}{\partial y} = 5xy e^{5xy};\) \(\frac{\partial F_2}{\partial z} = \cos(xz^3) - 3xz^3 \sin(xz^3)\)2. \(\frac{\partial F_3}{\partial x} = 5y^2 e^{5xy};\) \(\frac{\partial F_1}{\partial z} = x^2 y \cos(yz)\)3. \(\frac{\partial F_2}{\partial x} = -z^4 \sin(xz^3);\) \(\frac{\partial F_1}{\partial y} = x^2 z \cos(yz)\)Use these values to calculate:\(abla \times \mathbf{F} = (-3xz^3 \sin(xz^3) + 5xy e^{5xy}) \mathbf{i} - (5y^2 e^{5xy} - x^2 y \cos(yz)) \mathbf{j} - (z^4 \sin(xz^3) + x^2 z \cos(yz)) \mathbf{k}\).

Step 4: Calculating the Divergence

The divergence of a vector field \( \mathbf{F} = (F_1, F_2, F_3) \) is given by the scalar function:\[abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}\]Calculate each term separately:\(\frac{\partial F_1}{\partial x} = 2x \sin(yz); \) \( \frac{\partial F_2}{\partial y} = 0; \) \( \frac{\partial F_3}{\partial z} = 0 \).Thus, the divergence is:\(abla \cdot \mathbf{F} = 2x \sin(yz)\).

Key Concepts

Curl of a Vector Field

The curl of a vector field is a vector operation that helps us understand the rotation tendencies of the field at any point. When you have a vector field like \( \mathbf{F} = (F_1, F_2, F_3) \), the curl is concerned with how much and in what direction the field tends to swirl around a point. The formula for curl involves a determinant involving unit vectors and partial derivatives:\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_1 & F_2 & F_3 \end{vmatrix} \].

• The first component, in the direction of \( \mathbf{i} \), involves the partial derivatives \( \frac{\partial F_3}{\partial y} \) and \( \frac{\partial F_2}{\partial z} \).
• For \( \mathbf{j} \), it involves \( \frac{\partial F_3}{\partial x} \) and \( \frac{\partial F_1}{\partial z} \).
• The \( \mathbf{k} \) component involves \( \frac{\partial F_2}{\partial x} \) and \( \frac{\partial F_1}{\partial y} \).

To compute the curl, we take these partial derivatives and substitute them back into the equation. Bringing these together, the resulting vector gives insight into the swirling motion of the vector field. A non-zero curl at a point indicates a rotational effect, while a zero curl implies the field is more irrotational at that point.

Divergence of a Vector Field

Divergence is a concept that characterizes how much a vector field spreads out from a given point. Unlike curl, which focuses on rotation, divergence looks at expansion or contraction.For a vector field \( \mathbf{F} = (F_1, F_2, F_3) \), the divergence is calculated as a scalar:\[ abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \].It's essentially the sum of the field's partial derivatives with respect to each coordinate.

• A positive divergence indicates a source, meaning the field points outwards, spreading from the point.
• A negative divergence signifies a sink, showing that the field is converging towards the point.

In the provided example, the divergence is found to be \( 2x \sin(yz) \), which means the vector field's spreading tendencies vary with \( x \) and \( yz \). By evaluating divergence, we can understand how fluids or forces would behave in response to the given vector field, which is particularly useful in physics and engineering.

Partial Derivatives

Partial derivatives are crucial in vector calculus as they measure how a function changes as individual variables change, keeping others constant. In multivariable calculus, when you have a function like \( f(x, y, z) \), partial derivatives are found by differentiating with respect to one variable at a time.

• \( \frac{\partial f}{\partial x} \) indicates how \( f \) changes as \( x \) varies, with \( y \) and \( z \) held fixed.
• \( \frac{\partial f}{\partial y} \) and \( \frac{\partial f}{\partial z} \) work similarly for the other variables.

In the context of vector fields, partial derivatives help compute operations like curl and divergence. For the given vector field:

• The derivative \( \frac{\partial F_1}{\partial x} = 2x \sin(yz) \) shows how the field's first component changes with \( x \).
• Similarly, exploring these changes in \( y \) or \( z \) for other components unveils different aspects of the field's behavior.

By understanding partial derivatives, we can perform vector calculus operations more effectively, analyzing complex dynamic systems in mathematics, physics, and engineering.