Should be partial derivatives: ∂Fx/∂y = ∂Fy/∂xdaivinhtran said:A force F is conservative when :
dFx/dy = dFy/dx
They will be when you change to partial derivatives. (∂y/∂x = ∂x/∂y = 0)dFx/dy = d(x^2y^3)/dy = (2xdx/dy)(y^3) + (x^2)(3y^2)
dFy/dx = d(x^3/y^2)dx = (3x^2)(y^2) + (x^3)(2ydy/dx)
haruspex said:Should be partial derivatives: ∂Fx/∂y = ∂Fy/∂x
They will be when you change to partial derivatives. (∂y/∂x = ∂x/∂y = 0)
haruspex said:If you have a number of independent variables (in this case, x and y), and a function f of these, the partial derivative of f wrt x (written ∂f/∂x) means the change in f as x changes slightly but y stays constant. So when performing a partial derivative wrt x, treat y as a constant: ∂y/∂x = 0. Likewise ∂x/∂y = 0.
Plug those into the equation you got.
I don't know what education system you're in, let alone what's in what syllabus.daivinhtran said:Are those material in Calculus 2??
A conservative force is a type of force that does not dissipate energy as it moves an object. This means that the work done by a conservative force is independent of the path taken by the object and only depends on the initial and final positions of the object.
Some examples of conservative forces include gravity, electric forces, and magnetic forces. These forces can be represented by a potential energy function and follow the law of conservation of energy.
Unlike conservative forces, non-conservative forces dissipate energy as they move an object. This means that the work done by a non-conservative force is dependent on the path taken by the object and not just the initial and final positions.
Conservative forces and potential energy are closely related. The potential energy of an object in a conservative force field is equal to the negative of the work done by the force to move the object from a reference point to its current position.
Understanding conservative forces is essential in many areas of science and engineering. It allows for the calculation of potential energy and the prediction of an object's motion in a conservative force field. This knowledge is crucial in areas such as designing efficient machines and predicting the behavior of objects in space.