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Volume in Vector Fields 📂Vector Analysis

Volume in Vector Fields

Definition

The volume VV of a subspace D⊂RnD \subset \mathbb{R}^{n} in a Euclidean space is defined as follows when expressed in Cartesian coordinates u=(u1,u2,⋯ ,un)\textbf{u} = (u_{1}, u_{2}, \cdots , u_{n}).

V(D)=∫Ddu1du2⋯dun V(D) = \int_{D} du_{1} du_{2} \cdots d u_{n}

When u∈Rn\textbf{u} \in \mathbb{R}^{n} is transformed by a vector function f:Rn→Rn\textbf{f} : \mathbb{R}^{n} \to \mathbb{R}^{n} as f(u)=(f1(u),⋯ ,fn(u))\textbf{f} \left( \textbf{u} \right) = \left( f_{1} (\textbf{u}) , \cdots , f_{n} (\textbf{u}) \right), the volume of DD is as follows.

V(D)=∫D∣∂f(u)∂u∣du1du2⋯dun V(D) = \int_{D} \left| {{ \partial \textbf{f} (\textbf{u}) } \over { \partial \textbf{u} }} \right| d u_{1} d u_{2} \cdots d u_{n}

∣∂f(u)∂u∣\displaystyle \left| {{ \partial \textbf{f} (\textbf{u}) } \over { \partial \textbf{u} }} \right| represents the determinant of the Jacobian matrix of f(u)\textbf{f} (\textbf{u}), as can be shown. ∣∂f(u)∂u∣=det⁡[∂f1(u)∂u1⋯∂f1(u)∂un⋮⋱⋮∂fn(u)∂u1⋯∂fn(u)∂un] \left| {{ \partial \textbf{f} (\textbf{u}) } \over { \partial \textbf{u} }} \right| = \det \begin{bmatrix} {{\partial f_{1} (\textbf{u}) } \over {\partial u_{1} }} & \cdots & {{\partial f_{1} (\textbf{u}) } \over {\partial u_{n} }} \\ \vdots & \ddots & \vdots \\ {{\partial f_{n} (\textbf{u}) } \over {\partial u_{1} }} & \cdots & {{\partial f_{n} (\textbf{u}) } \over {\partial u_{n} }} \end{bmatrix}

Explanation

Some might feel intimidated just by seeing V(D)=∫D∣∂f(u)∂u∣du1⋯dun\displaystyle V(D) = \int_{D} \left| {{ \partial \textbf{f} (\textbf{u}) } \over { \partial \textbf{u} }} \right| d u_{1} \cdots d u_{n}. However, this post is meant to explain and make it easy to understand, so calmly read the explanation below.

Volume generalizes the concept of length in 11 dimensions, area in 22 dimensions, and volume in 33 dimensions to n∈Nn \in \mathbb{N} dimensions. In Korea, volume is typically translated to ‘volume’ in English, but in mathematics, these concepts are not distinctly categorized, hence using the term [volume] as it is pronounced, might evoke the concept of n=3n=3 dimensions due to generalization.

Coordinate Transformation

The vector function f\textbf{f} can have various applications depending on the field, but if thought of in the context of physics, it can be considered as a transformation of coordinate systems. For instance, if f\textbf{f} is given as follows:

f(r,θ)=(f1(r,θ),f2(r,θ))=(x(r,θ),y(r,θ))=(rcos⁡θ,rsin⁡θ) \begin{align*} \textbf{f} (r,\theta) =& \left( f_{1} (r,\theta) , f_{2} (r,\theta) \right) \\ =& \left( x (r,\theta) , y (r,\theta) \right) \\ =& (r \cos \theta , r \sin \theta) \end{align*}

This becomes the polar coordinate system. Eliminating f\textbf{f} and converting x=x(r,θ)x = x (r,\theta) and y=y(r,θ)y = y (r,\theta) into expressions familiar to us:

x=rcos⁡θy=rsin⁡θ x = r \cos \theta \\ y = r \sin \theta

The determinant of this Jacobian matrix is:

det⁡[∂x∂r∂x∂θ∂y∂r∂y∂θ]=det⁡[cos⁡θsin⁡θ−rsin⁡θrcos⁡θ]=rcos⁡2θ+rsin⁡2θ=r \begin{align*} \det \begin{bmatrix} {{\partial x } \over {\partial r }} & {{\partial x } \over {\partial \theta }} \\ {{\partial y } \over {\partial r }} & {{\partial y } \over {\partial \theta }} \end{bmatrix} =& \det \begin{bmatrix} \cos \theta & \sin \theta \\ -r \sin \theta & r \cos \theta \end{bmatrix} \\ =& r \cos^{2} \theta + r \sin^{2} \theta \\ =& r \end{align*}

Therefore, the area (volume) V(R)V(R) of the given region R⊂R2R \subset \mathbb{R}^{2} in 22 dimensions is calculated as follows:

V(R)=∫Rrdrdθ V(R) = \int_{R} r dr d\theta

It is important not to confuse that f\textbf{f} does not map (x,y)(x,y) to (r,θ)(r,\theta); instead, we used the point (r,θ)(r, \theta) in the polar coordinate system to represent the point (x,y)(x,y) in the Cartesian coordinate system, meaning f\textbf{f} maps (r,θ)(r, \theta) to (x,y)(x,y).

Why is VV Defined That Way

To explain why the volume VV is defined in such a way, it’s better to look at the process of how volume is derived. Let’s start with the infinitesimal volume. Generally, in dimensions lower than 33, the following dx,dA,dVdx, dA, dV are sequentially called infinitesimal length, infinitesimal area, and infinitesimal volume.

dx=dxdA=dxdydV=dxdydz \begin{align*} dx =& dx \\ dA =& dxdy \\ dV =& dxdydz \end{align*}

Just as the length of the 11 dimensional interval I=[x1,x2]I = [x_{1},x_{2}] can be calculated as:

(x2−x1)=∫x1x2dx=∫Idx (x_{2} - x_{1}) = \int_{x_{1}}^{x_{2}} dx = \int_{I} dx

the area of a 22 dimensional rectangle R=[x1,x2]×[y1,y2]R = [x_{1}, x_{2}] \times [y_{1} , y_{2}] is:

(x2−x1)(y2−y1)=∫y1y2∫x1x2dxdy=∫RdA (x_{2} - x_{1}) (y_{2} - y_{1}) = \int_{y_{1}}^{y_{2}} \int_{x_{1}}^{x_{2}} dxdy = \int_{R} dA

and the volume of a 33 dimensional rectangular parallelepiped D=[x1,x2]×[y1,y2]×[z1,z2]D = [x_{1}, x_{2}] \times [y_{1} , y_{2}] \times [z_{1} , z_{2}] is:

(x2−x1)(y2−y1)(z2−z1)=∫z1z2∫y1y2∫x1x2dxdydz=∫DdV (x_{2} - x_{1}) (y_{2} - y_{1}) (z_{2} - z_{1}) = \int_{z_{1}}^{z_{2}} \int_{y_{1}}^{y_{2}} \int_{x_{1}}^{x_{2}} dxdydz = \int_{D} dV

When generalizing the process of calculating volume in Cartesian coordinates, the infinitesimal volume

dV=du1du2⋯dun dV = d u_{1} d u_{2} \cdots d u_{n}

and applying the definite integral ∫D\displaystyle \int_{D} to both sides of the equation, one can accept the following expression: ∫Ddu1du2⋯dun=∫DdV=V(D) \int_{D} d u_{1} d u_{2} \cdots d u_{n} = \int_{D} dV = V(D)

Here, DD doesn’t need to necessarily be the Cartesian product of closed intervals, and its shape could be round, star-shaped, or anything else. However, the actual calculation might be challenging, and that’s when coordinate transformation comes handy, making such mathematical formulations easier to handle.