# Volume element in different coordinate system

• teddd
In summary: This gives you the correct answer of dV= rdrd\theta.In summary, when changing coordinate systems, the infinitesimal volume element can be transformed by multiplying by the determinant of the Jacobian of the transformation. However, when dealing with differentials, this process involves using the "skew product" or cross product. In the example of transforming from cartesian to polar coordinates, the correct result is obtained by taking the determinant of the Jacobian, r, and multiplying it by the differentials dr and d\theta.
teddd
Very simple question:

Let $x^0,x^1,...,x^n$ be some fixed coordinate system, so that the infinitesimal volume element is $dV=dx^0dx^1...dx^n$.
Then any change to a new (primed) coordinate system $x^{0'},x^{1'},...,x^{n'}$ transforms the volume to $$dV=\frac{\partial (x^0,x^1,...,x^n)}{\partial (x^{0'},x^{1'},...,x^{n'})}dx^{0'}dx^{1'}...dx^{n'}$$ where $\frac{\partial (x^0,x^1,...,x^n)}{\partial (x^{0'},x^{1'},...,x^{n'})}$ is the determinat of the jacobian of the transformation.

So let's try to do this in a concrete example: the transformation from cartesian $x,y$ to polar $r,\theta$ coordinates.
The Jacobian is simply $r$ and so i get to $dV=dxdy=rdrd\theta$.

Doing the math i get $$dV^{pol.}=r(\cos\theta dx+\sin\theta dy)(-\frac{1}{r}\sin\theta dx+\frac{1}{r}\cos\theta dy)$$since $dr=\cos\theta dx+\sin\theta dy$ and $d\theta=-\frac{1}{r}\sin\theta dx+\frac{1}{r}\cos\theta dy$.
But then:$$dV^{pol.}=-\cos\theta\sin\theta dx^2+\cos^2\theta dxdy-\sin^2\theta dxdy + \cos\theta\sin\theta dy^2$$ and is clear that $dV^{cart.}\neq dV^{pol.}$ !
Where is my mistake?? Thanks for your help guys!

Last edited:
You can't just multiply differentials.

Try using the formal rules

dx dx = dy dy = 0

and

dx dy = -dy dx

$$x= r cos(\theta)$$ $$y= r sin(\theta)$$

$$dx= cos(\theta)dr- r sin(\theta)d\theta$$

$$dy= sin(\theta)dr+ r cos(\theta)d\theta$$

But when you multiply differentials, you have to use "skew product" Petr Mugver mentions.

A good way of thinking about it is as vectors
$$dx= cos(\theta)dr\vec{i}- r sin(\theta)d\theta\vec{j}$$
$$dy= sin(\theta)dr\vec{i}+ r sin(\theta)d\theta\vec{j}$$

Now the product is the cross product of the two "vectors".

$$\left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ cos(\theta) & - rsin(\theta) & 0 \\ sin(\theta) & rcos(\theta) & 0 \end{array}\right|= (rcos^2(\theta)+ rsin^2(\theta))drd\theta\vec{k}= r drd\theta \vec{k}$$

## 1. What is a volume element in different coordinate systems?

A volume element is a small unit of volume that is used to measure the size or capacity of a three-dimensional object or space. In different coordinate systems, the volume element may be represented by different mathematical expressions or symbols, but it still serves the same purpose of measuring volume.

## 2. How is the volume element calculated in rectangular coordinates?

In rectangular coordinates, the volume element is calculated by multiplying the length, width, and height of the object. The formula for calculating the volume element in rectangular coordinates is V = l x w x h, where V is the volume, l is the length, w is the width, and h is the height.

## 3. Can the volume element be negative in some coordinate systems?

Yes, in some coordinate systems, the volume element can be negative. This usually occurs in non-Cartesian coordinate systems, such as spherical or cylindrical coordinates, where the orientation of the axes can result in negative values for the volume element.

## 4. How does the volume element change in different coordinate systems?

The volume element changes in different coordinate systems because the mathematical expressions used to calculate it are based on the specific characteristics and orientation of the coordinate system. For example, in spherical coordinates, the volume element is calculated using the radius, colatitude, and azimuth angles, whereas in cylindrical coordinates, it is calculated using the radius, height, and angle of rotation.

## 5. Why is it important to consider the volume element in different coordinate systems?

It is important to consider the volume element in different coordinate systems because it allows us to accurately measure the volume of objects or spaces in a variety of orientations and shapes. It also helps in solving mathematical problems and equations involving volume in different coordinate systems, such as in physics, engineering, and other scientific fields.

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