# Proving F = grad(f) with 3 Variables

• jwxie
In summary, the condition for a vector field in three-space to be the gradient of some scalar field is that it must have zero curl, which can be expressed as three equations between the partial derivatives of its components. The proof for this fact is similar to the one used in the plane case, using the concept of a potential function and Stokes's theorem.
jwxie

## Homework Statement

For two variables, F = grad(f) if only if partial P / partial y is equal to partial Q / partial x
where, P and Q represent the x, y function, P(x,y) and Q(x,y)

For example:
F(x,y) = P(x,y) + Q(x,y)

Now the question is, for three variables, I have P + Q + R.
Is proving partial P / partial y = partial Q / partial x enough to say that F = grad(f)? If not, what is the approach to prove it for a 3 variables?

Thank you

The condition for a vector field in three-space to be the gradient of some scalar field is a generalization of the condition in the plane: the vector field must have zero curl, $$\nabla\times\mathbf{F} = 0$$. This works out to three equations, not one, between the partial derivatives of the components of $$\mathbf{F}$$: $$\frac{\partial F_x}{\partial y} - \frac{\partial F_y}{\partial x} = \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} = \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = 0$$.

The proof of this fact uses the same argument as in the plane case (a gradient vector field is one whose integral around a closed loop is always zero, so you can construct a potential function $$f$$ with $$\mathbf{F} = \nabla f$$ by integration) together with Stokes's theorem.

Thank you very much!

## 1. What is the meaning of F = grad(f) in 3 variables?

F = grad(f) is a mathematical notation that represents the relationship between a scalar function f and its gradient in three-dimensional space. It is used to show that the vector field F is equal to the gradient of the scalar function f.

## 2. How do you prove F = grad(f) with 3 variables?

To prove that F = grad(f) with 3 variables, you need to show that the components of the vector field F are equal to the partial derivatives of the scalar function f with respect to each variable. This can be done using the definition of the gradient and the rules of vector calculus.

## 3. What are the applications of proving F = grad(f) with 3 variables?

The proof of F = grad(f) with 3 variables is useful in various fields of science and engineering, such as fluid dynamics, thermodynamics, and electromagnetism. It allows for the calculation of important physical quantities, such as potential energy, electric field, and heat transfer, which are essential in understanding and predicting the behavior of complex systems.

## 4. Is it necessary to prove F = grad(f) with 3 variables in every case?

In some cases, it is not necessary to explicitly prove F = grad(f) with 3 variables. This is because certain vector fields can be easily identified as the gradient of a scalar function. However, in more complex systems, it is important to prove this relationship in order to accurately analyze and model the behavior of the system.

## 5. Can F = grad(f) be extended to more than 3 variables?

Yes, F = grad(f) can be extended to any number of variables. The notation F = grad(f) is commonly used in three-dimensional space, but it can also be applied to higher dimensions. In fact, the concept of the gradient can be extended to vector fields with any number of dimensions, making it a powerful tool in mathematical analysis and physics.

Replies
5
Views
1K
Replies
4
Views
1K
Replies
4
Views
531
Replies
1
Views
1K
Replies
18
Views
2K
Replies
8
Views
1K
Replies
9
Views
973
Replies
2
Views
980
Replies
1
Views
1K
Replies
21
Views
2K