# Conservative Vector Field Potential

• tazzzdo
In summary, the given vector field \vec{E}(\vec{r}) = \vec{r}/r^2 is conservative, and its scalar potential is f = log r. This was determined by showing that \vec{E} is the gradient of a scalar function, using implicit differentiation and solving for f. This solution was presented in a Vector Calculus class as an extra credit question.
tazzzdo

## Homework Statement

Let $\vec{E}$($\vec{r}$) = $\vec{r}$/r2, r = |$\vec{r}$|, $\vec{r}$ = x$\hat{i}$ + y$\hat{j}$ + z$\hat{k}$ be a vector field in ℝ3. Show that $\vec{E}$ is conservative and find its scalar potential.

## Homework Equations

All of the above.

## The Attempt at a Solution

$\vec{\nabla}$ $\times$ $\vec{E}$ = $\vec{0}$ $\Rightarrow$ $\vec{E}$ is conservative $\Rightarrow$ $\vec{E}$ = $\vec{\nabla}$f, where f is a scalar function.

xf = x/r2
yf = y/r2
zf = z/r2

r2 = x2 + y2 + z2

By implicit differentiation:

2r$\frac{∂r}{∂x}$ = 2x $\Rightarrow$ $\frac{∂r}{∂x}$ = x/r

And as follows:

$\frac{∂r}{∂y}$ = y/r, $\frac{∂r}{∂z}$ = z/r

$\frac{x}{r}$ $\cdot$ $\frac{1}{r}$ = r-1$\frac{x}{r}$ = r-1$\frac{∂r}{∂x}$ = $\frac{∂}{∂x}$(log r)

$\Rightarrow$ f = log r

This is an extra credit question in my Vector Calculus class. I think this is the correct solution, but I'm not positive.

It looks fine to me.

## 1. What is a conservative vector field potential?

A conservative vector field potential is a scalar function that is associated with a conservative vector field. It represents the work done by the conservative vector field in moving an object from one point to another, and is independent of the path taken.

## 2. How is a conservative vector field potential different from a non-conservative vector field potential?

A conservative vector field potential is a type of potential that is associated with a conservative vector field, meaning that the work done by the field is independent of the path taken. On the other hand, a non-conservative vector field potential is associated with a non-conservative vector field, meaning that the work done by the field is dependent on the path taken.

## 3. What are some real-world applications of conservative vector field potential?

Conservative vector field potential has many applications in physics, particularly in the study of forces and motion. Examples include the gravitational potential in celestial mechanics, the electric potential in electrostatics, and the magnetic potential in electromagnetism.

## 4. How can conservative vector field potential be calculated?

Conservative vector field potential can be calculated using a method called line integration, where the potential is determined by integrating the field along a given path. Alternatively, it can also be calculated using a mathematical expression known as the gradient, which represents the rate of change of the potential with respect to position.

## 5. What are some properties of conservative vector field potential?

Some properties of conservative vector field potential include: it is a scalar function, it is path-independent, it is defined in terms of a vector field, and it satisfies the Laplace's equation. Additionally, the potential is related to the work done by the field, and the gradient of the potential provides the direction of the field at any given point.

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