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Definition/Summary
Given a vector field ##\vec F(x,y,z)## that has a potential function, how do you find it?
Equations
$$\nabla \phi(x,y,z) = \vec F(x,y,z)$$ $$\nabla \times \vec F(x,y,z) = \vec 0$$
Extended explanation
Suppose we are given a vector field ##\vec F(x,y,z)=\langle f(x,y,z),g(x,y,z),h(x,y,z)\rangle## that has a potential function ##\phi## and we wish to recover the potential function. We know that we must have##\nabla \phi =\vec F##, so ##\phi_x = f,\, \phi_y=g,\, \phi_z = h##. This means we can recover ##\phi## by integrating the components of ##\vec F##.
To look at a particular example, consider $$\vec F =\langle 2xz^3+e^z,-z\sin(yz),3x^2z^2-y\sin(yz)+xe^z\rangle$$Our unknown potential function ##\phi## must satisfy$$\phi_x=2xz^3+e^z,\, \phi_y=-z\sin(yz),\,\phi_z=3x^2z^2-y\sin(yz)+xe^z$$Students often solve this type of problem by taking the anti-partial derivative of each equation:$$\phi = \int 2xz^3+e^z\,\partial x = x^2z^3+xe^z$$ $$\phi = \int -z\sin(yz)\...
Potential functions are mathematical functions that can be used to model and describe a physical system. They are often used in physics, engineering, and other sciences to analyze the behavior of a system and make predictions.
A function is a potential function if it satisfies the condition of being a conservative vector field. This means that the function's derivative, also known as the gradient, must be equal to zero. In other words, the function must have a constant value along any path within its domain.
Finding potential functions is important because it allows us to simplify complex physical systems and make accurate predictions about their behavior. It also helps us understand the underlying principles and relationships within a system.
Some common techniques for finding potential functions include using the gradient, finding the line integral, and using the method of partial fractions. These techniques involve manipulating the function and its derivatives to satisfy the condition of being a conservative vector field.
Potential functions can be used for many different types of systems, including mechanical, electrical, and chemical systems. However, not all systems can be described by potential functions, and in some cases, other mathematical models may be more appropriate.