How Do You Calculate Force from a Two-Dimensional Potential Energy Function?

[Thermon]

Homework Statement

A potential energy function for a two-dimensional force is of the form U = 3x³ * y - 7x.
Find the force that acts at the point (x, y).[/B]

Homework Equations

In a 1-dimensional case:
ΔU = -∫Fx dx
dU = -Fx dx
Fx = -dU/dx

The Attempt at a Solution

I know how to find the force in a 1-dimensional case; it's the gradient at the given x.

But I can't wrap my head around it when there are two variables.

Could it perhaps be the sum of the derivatives; Fx = -(dU/dx + dU/dy)?[/B]

 

[LCKurtz]

Homework Statement

A potential energy function for a two-dimensional force is of the form U = 3x³ * y - 7x.
Find the force that acts at the point (x, y).[/B]

Homework Equations

In a 1-dimensional case:
ΔU = -∫Fx dx
dU = -Fx dx
Fx = -dU/dx

The Attempt at a Solution

I know how to find the force in a 1-dimensional case; it's the gradient at the given x.

But I can't wrap my head around it when there are two variables.

Could it perhaps be the sum of the derivatives; Fx = -(dU/dx + dU/dy)?[/B]

The force will be a vector. If $\phi(x,y)$ is a potential for $\vec F(x,y)$ then$$
\vec F(x,y) = \langle \phi_x,\phi_y\rangle$$

 

[Thermon]

The force will be a vector. If $\phi(x,y)$ is a potential for $\vec F(x,y)$ then$$
\vec F(x,y) = \langle \phi_x,\phi_y\rangle$$

So it's the combined vector of V_y and V_x? Correct me if I'm wrong, but does that means that the force at (x, y) would be the net vector of the E_pot up and downwards against E_kin?
Finding those two would involve finding the integral of both F_x and F_y

 

[LCKurtz]

The force will be a vector. If $\phi(x,y)$ is a potential for $\vec F(x,y)$ then$$
\vec F(x,y) = \langle \phi_x,\phi_y\rangle$$

So it's the combined vector of V_y and V_x? Correct me if I'm wrong, but does that means that the force at (x, y) would be the net vector of the E_pot up and downwards against E_kin?
Finding those two would involve finding the integral of both F_x and F_y

I don't know what you mean by "combined vector" and "net vector" and "upwards and downwards". It is a force vector field having two components or a magnitude and direction. And I don't know what F_x and F_y you are talking about. You get the vector field from the potential by taking the partials of the potential, not integrating, as I gave in the formula above.