Exercise 12

In this exercise, we will look at how different forms of component function's dependence on our variables will change the curl of the vector field. For each of the vector fields given below, decide if the curl of the vector field must be zero (as a vector) and write a sentence or two to justify your decision.

1. $$\begin{gathered} \vF=f(x)\vi+g(y)\vk+\vj+h(z)\vj \end{gathered}$$

2. $$\begin{gathered} \vG=f(x)\vi+f(y)\vk+\vj+f(z)\vj \end{gathered}$$

3. $$\begin{gathered} \vH=f(y,z)\vi+g(x,z)\vk+\vj+h(x,y)\vj \end{gathered}$$

4. $$\begin{gathered} \vA=g(x,y)\vi-g(x,y)\vk+\vj+h(z)\vj \end{gathered}$$

5. $$\begin{gathered} \vB=g(x,y)\vi-g(y,x)\vk+\vj+h(z)\vj \end{gathered}$$