logo

Definition and Properties of Vector Areas 📂Mathematical Physics

Definition and Properties of Vector Areas

Definition

vector_area.png

For a given surface $S$, the following integral is called the vector area of $S$.

$$ \mathbf{a} := \int_{\mathcal{S}} d \mathbf{a} $$

Description

hemisphere.png

As an example, let’s calculate the vector area of a hemisphere with a radius of $R$. It is $d \mathbf{a} = R^{2}\sin\theta d\theta d\phi \hat{\mathbf{r}}$. Here,

$$ \hat{\mathbf{r}} = \cos\phi \sin\theta \hat{\mathbf{x}} + \sin\phi \sin\theta\hat{\mathbf{y}} + \cos\theta\hat{\mathbf{z}} $$

when integrated over the region of the northern hemisphere, both $\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$ components cancel out, leaving only the $\hat{\mathbf{z}}$ component. Thus, we obtain the following.

$$ \begin{align*} \mathbf{a} &= \int_{\phi=0}^{2\pi} \int_{\theta=0}^{\pi/2} R^{2}\sin\theta \cos\theta d\theta d\phi \hat{\mathbf{z}} \\ &= 2\pi R^{2} \int_{\theta=0}^{\pi/2} \sin\theta \cos\theta d\theta \hat{\mathbf{z}} \\ &= 2\pi R^{2} \dfrac{1}{2} \hat{\mathbf{z}} \\ &= \pi R^{2} \hat{\mathbf{z}} \end{align*} $$

The integral on $\theta$ is justified by the table of integrals of trigonometric functions at $(1)$.

Properties

  1. The vector area of a closed surface is always $\mathbf{a} = \mathbf{0}$.

  2. The vector area of surfaces with the same boundary is always the same.

  3. The following integral holds. $$ \mathbf{a} = \dfrac{1}{2}\oint \mathbf{r} \times d \mathbf{l} $$

  4. For any constant vector $\mathbf{c}$, the following is true. $$ \oint (\mathbf{c} \cdot \mathbf{r}) d \mathbf{l} = \mathbf{a} \times \mathbf{c} $$