Question

Find curl \(F\) for the vector field at the given point. $$ \mathbf{F}(x, y, z)=e^{-x y z}(\mathbf{i}+\mathbf{j}+\mathbf{k}) $$

Short answer

The curl of the vector field \(F(x, y, z)=e^{-x y z}(\mathbf{i}+\mathbf{j}+\mathbf{k})\) is \(\nabla \times F = \mathbf{0}\).

Step-by-step solution

Take the necessary derivatives

The first step would involve taking the partial derivatives in the expression for the curl of F. Starting with the required derivatives for the x-component of the curl, \(\frac{\partial C}{\partial y} = \frac{\partial (e^{-xyz})}{\partial y} = -xze^{-xyz} \) and \(\frac{\partial B}{\partial z} = \frac{\partial (e^{-xyz})}{\partial z} = -xye^{-xyz}\). Similarly, the required derivatives for the y-component and z-component can also be obtained. They all turn out to be \(-xye^{-xyz}\), \(-xye^{-xyz}\), and \(-xze^{-xyz}\) respectively.

Substitute and simplify

Substuituting these derivatives into the expression for curl gives \( \nabla \times F = \left(-xze^{-xyz} +xye^{-xyz} \right) \mathbf{i} - \left(-xye^{-xyz} + xye^{-xyz} \right) \mathbf{j} + \left(-xye^{-xyz} + xye^{-xyz} \right) \mathbf{k}\). In every component the same term is added and subtracted, so when simplified, we actually get \( \nabla \times F = 0 \mathbf{i} - 0 \mathbf{j} + 0 \mathbf{k}\)

Write out the answer in vector form

The components of the vector are all zero, so in vector notation just write out \( \nabla \times F = \mathbf{0}\)