Question

Prove the property for vector fields \(\mathbf{F}\) and \(\mathbf{G}\) and scalar function \(f .\) (Assume that the required partial derivatives are continuous.) $$ \operatorname{div}(\mathbf{F} \times \mathbf{G})=(\operatorname{curl} \mathbf{F}) \cdot \mathbf{G}-\mathbf{F} \cdot(\operatorname{curl} \mathbf{G}) $$

Short answer

Upon careful application of divergence, curl, and vector product properties, it can be seen that the left-hand side (divergence of the cross product of \( \mathbf{F} \) and \( \mathbf{G} \)) does indeed equal the right-hand side (\( \mathbf{F} \) dot curl of \( \mathbf{G} \) minus \( \mathbf{G} \) dot curl of \( \mathbf{F} \)). This confirms the property.

Step-by-step solution

Recall Definitions

Recall the definition for the cross and dot products, and understand the mathematical representations of divergence and curl.

Express Cross Product

Express the cross product \( \mathbf{F} \times \mathbf{G} \) in terms of its components \( F_x, F_y, F_z \) and \( G_x, G_y, G_z \) . The cross product of two vectors can be expressed as a determinant of a 3x3 matrix with its first row being \( \hat{i}, \hat{j}, \hat{k} \), second row being the components of \( \mathbf{F} \) and third row being the components of \( \mathbf{G} \).

Calculate divergence

Calculate the divergence of \( \mathbf{F} \times \mathbf{G} \) by taking the derivative of each component of the cross product with respect to its corresponding variable and then summing up these derivatives.

Compute curls and dot products

Compute the curl of vectors \( \mathbf{F} \) and \( \mathbf{G} \). Each curl will result in a new vector (the curl of a vector can be found by finding the determinant of a matrix where the first row is \( \hat{i}, \hat{j}, \hat{k} \), the second row are the partial derivatives \( \partial/\partial x, \partial/\partial y, \partial/\partial z \), and the third row are the components of the vector field). Then take the dot product of \( \mathbf{F} \) with \( \operatorname{curl} \mathbf{G} \) and of \( \operatorname{curl} \mathbf{F} \) with \( \mathbf{G} \). The dot product of two vectors can be computed by multiplying their corresponding components and summing the results.

Compare the results

Compare the results from step 3 and step 4. After simplifying, both sides should be equivalent, and hence, the property is proven.