Open study

is now brainly

With Brainly you can:

  • Get homework help from millions of students and moderators
  • Learn how to solve problems with step-by-step explanations
  • Share your knowledge and earn points by helping other students
  • Learn anywhere, anytime with the Brainly app!

A community for students.

Find a potential function for: F = y sin(z)i + x sin(z)j + xy cos(z)k

Mathematics
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Join Brainly to access

this expert answer

SEE EXPERT ANSWER

To see the expert answer you'll need to create a free account at Brainly

By "potential function" you mean some function \(f(x,y,z)\) of three variables such that \[\frac{\partial}{\partial x}f(x,y,z)=y\sin(z)\]\[\frac{\partial}{\partial y}f(x,y,z)=x\sin(z)\]and\[\frac{\partial}{\partial z}f(x,y,z)=xy\cos(z)\]correct?
The potential of F is any function f such that del f = F.
Good to know. Then, to solve this, notice that \[\frac{\partial}{\partial x}f(x,y,z)=y\sin(z)\]is a constant function of \(x\). Likewise, \[\frac{\partial}{\partial y}f(x,y,z)=x\sin(z)\]is a constant function of \(y\). What does that tell you about what the function \(f(x,y,z)\) has to look like?

Not the answer you are looking for?

Search for more explanations.

Ask your own question

Other answers:

I got this: xycos(z), xycos(z), -xysin(z). Is this correct?
Don't quote me on this, but I believe George gave you part of the answer. Now if I remember correctly all you have to do integrate each part of his answer with respect to x, y, and z and then find the constant of integration. That is, intergrate the first part with respect to x, the second part with respect to y, and the last part with respect to z. Then find the constant of integration. I can't do it for you because I am not really sure how to go about it. I'ts been over 10 years since I graduate it from engineering school.
There is a methodical way of doing this.
I did the integration part and got the above answer but how do I find the constant of integration?
\[ \frac{\partial}{\partial x}f(x,y,z)=y\sin(z) \implies f(x, y, z) = xy\sin(z)+g(y, z) \]
So then you integrate another function to solve for \(g(y, z)\).
Since you're asked for "a" potential function, you don't need to worry about a general constant C, since any constant will work, you can just choose 0 and be done with it.
If I remember correctly I think you can equate all our integrals to each other and then compare coefficients. May be you would be better off asking one of the resident geniuses.
From what I'm seeing, wio's way of doing this will get you to a solution, rather easily.
\[ \frac{\partial}{\partial y} xy\sin(z) +g(y, z) =x \sin(z) +g'(y, z) \]\[ \frac{\partial}{\partial y}f(x,y,z)=x\sin(z) \]This tells us what?
It seems that \(g'(y, z) = 0\). So we know that \(g(y, z)\) is a constant with respect to \(y\).
wonder why none of the local residents just give you the answer
^^because giving answers for free is against the Code of Conduct.
So we have: \[ f(x, y, z) =xy\sin(z) +h(z) \]\[ \frac{\partial}{\partial z} xy\sin(z) +h(z) = xy\cos(z) + h'(z) \]\[ \frac{\partial}{\partial z}f(x,y,z)=xy\cos(z) \]So \(h'(z)=0\). It's pretty obvious now that our potential function is just: \[ f(x, y, z) = xy\sin(z) +C \]And as @KingGeorge said, the \(C\) will work for any constant. In fact there just isn't anyway for us to know what \(C\) is.
what is "giving the answer for free?"
@blackjesus It's giving them a solution without having them put forth any effort on their own part to solve it.
^^well said.
Ideally you walk the through it. Sometimes people are so confused that they need a walk through of a problem to understand the method.
Oh! Without "giving" the answer away, if you take wio's answer and find the gradient of it. If you end up with the function F that you started with. You have the right answer.
Addendum: The only way to learn Math is to do lots of problems. However, answering a question with a question will not help the student learn anything. I have to go now. Good bye. Good luck.
Yes, the methodology is as follows: 1) Integrate the function in terms of \(x\) (or some other variable). \( f(x, y, z) = \int F_xdx +g(y, z) \) 2) Take the derivative in terms of \(y\) (or some other variable). Then solve for \(g'(y, z)\). \(F_y = \frac{\partial }{\partial y}\int F_xdx +g'(y, z)\) \(g'(y, z) = F_y - \frac{\partial }{\partial y}\int F_xdx \) 3) Integrate in terms of \(y\). \(\int g'(y, z)dy = g(y, z) + h(z) \) 4) Solve for \(h(z)\) the same way we did for \(g(y, z)\).

Not the answer you are looking for?

Search for more explanations.

Ask your own question