1. The problem statement, all variables and given/known data Find a function f(x,y,z) such that F = (gradient of F). 3. The attempt at a solution I don't know :( I'm so confused Please help me!!
hi calculusisrad! do you mean "Find a function f(x,y,z) such that F = (gradient of f)" ? (only scalars have gradients, there's no gradient of a vector) i don't understand either is either f or F given in the question?
Think about what a gradient is. If I told you to find the gradient of a function, what would you do? You would differentiate the function wrt x, and that is the i component of the gradient, you would differentiate the function wrt y, and that is the j component, and then you would differentiate the function wrt z, and that is the k component. Now, we are going in reverse. What is the reverse of differentiation?
I completely forgot the biggest part of the problem. WOW. Sorry about that!!! Let F = (2xye^z)i + ((e^z)(x^2))j + ((x^2)y(e^z)+(z^2))k NOW find a function f(x,y,z) such that F = Gradient of f. Sorry about that. Please answer :)
It's pretty easy to guess a form for f that works. Start guessing. That's often the easiest way to solve problems like this. What's a likely form for f given the first component of F?
You know what the definition of "gradient" is, so use that. [itex]\frac{\partial f}{\partial x}=[/itex] what? [itex]\frac{\partial f}{\partial y}=[/itex] what? [itex]\frac{\partial f}{\partial z}=[/itex] what?