Question

Explain how to compute the divergence of the vector field \(\mathbf{F}=\langle f, g, h\rangle\).

Short answer

Answer: The formula used to calculate the divergence of a vector field \(\mathbf{F} = \langle f, g, h\rangle\) is \(\nabla \cdot \mathbf{F} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} + \frac{\partial h}{\partial z}\), where \(\frac{\partial f}{\partial x}\), \(\frac{\partial g}{\partial y}\), and \(\frac{\partial h}{\partial z}\) are partial derivatives of the components \(f\), \(g\), and \(h\) with respect to their respective variables.

Step-by-step solution

Determine the components of the vector field

The given vector field \(\mathbf{F}\) can be represented as: $$ \mathbf{F} = \langle f, g, h\rangle$$ Here, \(f\), \(g\), and \(h\) are the components of the vector field in the \(x\), \(y\), and \(z\) directions, respectively.

Understand the divergence formula

The divergence of a vector field \(\mathbf{F}\) is denoted as \(\nabla \cdot \mathbf{F}\) and is calculated using the following formula: $$ \nabla \cdot \mathbf{F} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} + \frac{\partial h}{\partial z}$$ Here, \(\frac{\partial f}{\partial x}\), \(\frac{\partial g}{\partial y}\), and \(\frac{\partial h}{\partial z}\) are partial derivatives of the components \(f\), \(g\), and \(h\) with respect to their respective variables.

Calculate the partial derivatives

Using the divergence formula, we need to find the partial derivatives: 1. \(\frac{\partial f}{\partial x}\): Partial derivative of \(f\) with respect to \(x\). 2. \(\frac{\partial g}{\partial y}\): Partial derivative of \(g\) with respect to \(y\). 3. \(\frac{\partial h}{\partial z}\): Partial derivative of \(h\) with respect to \(z\).

Calculate the divergence

After calculating the partial derivatives in Step 3, substitute their values into the divergence formula: $$ \nabla \cdot \mathbf{F} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} + \frac{\partial h}{\partial z}$$ and simplify the expression to find the divergence of the vector field \(\mathbf{F}\).