Definite Integrals of Vector Fields

Integrals over Curves

Definition: Integrals over Parametric Curves

Let $\mathbf{F}:\mathcal{D}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ be a Real Vector Field with component functions $f_1,\dots ,f_n$ and let $\gamma :[a;b]\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be a Parametric Curve with such that $\gamma ([a;b])\subseteq \mathcal{D}$.

The integral of $\mathbf{F}$ over $\gamma$ is defined as the vector whose components are the line integrals of $f_1,\dots ,f_n$ over $\gamma$:

$${\int }_{\gamma }\mathbf{F}\stackrel{\text{def}}{=}\left[\begin{array}{c}{\int }_{\gamma }{f}_1\\ ⋮\\ {\int }_{\gamma }{f}_n\end{array}\right]$$

It is also possible to define an integral of a Real Vector Field over a curve which depends only on its geometry and on the way it is traversed by a particular parametrization using the following theorem.

Theorem: Integrals over Equivalent Parametrizations

Let $\mathbf{F}:\mathcal{D}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ be a Real Vector Field, let $\mathcal{C}\subseteq \mathcal{D}$ be a curve in ${\mathbb{R}}^n$ and let $\gamma :[a;b]\subseteq \mathbb{R}\to {\mathbb{R}}^n$ and $\phi :[c;d]\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be parametrizations of $\mathcal{C}$.

If $\gamma$ and $\phi$ are continuously differentiable on $(a;b)$ and $(c;d)$, respectively, and are equivalent up to a continuously differentiable reparametrization, then the integrals of $\mathbf{F}$ over $\gamma$ and $\phi$ are equal:

$${\int }_{\gamma }\mathbf{F}={\int }_{\phi }\mathbf{F}$$

PROOF

This is guaranteed by applying the corresponding theorem for scalar line integrals for each of the component functions.

The aforementioned theorem guarantees that the integrals of a vector field $\mathbf{F}$ over continuously differentiable parametrizations of the same curve $\mathcal{C}$ which are equivalent up to a continuously differentiable reparametrization are equal. However, while some parametrizations may be equivalent amongst each other and some other parametrizations may also be equivalent amongst each other, this does not ensure that these groups of parametrizations are all equivalent. If we want to define the notion of an integral over a curve geometrically, we need to determine which equivalence class to consider. A very natural choice is based on the equivalence of regular injective parametrizations. According to the above theorem, the integrals of $\mathbf{F}$ over these parametrizations are all equal and, since they are also injective, this value depends only on the length of $\mathcal{C}$.

Definition: Vector Field Integrals over Curves

The integral of a Real Vector Field $\mathbf{F}$ over a curve $\mathcal{C}$ is defined as the integral of $\mathbf{F}$ over any continuously differentiable parametrization $\gamma$ of $\mathcal{C}$ with a non-vanishing derivative

$${\int }_{\gamma }\mathbf{F}$$

NOTATION

$${\int }_{\mathcal{C}}\mathbf{F}$$

Warning: Integrals over Curves vs Vector Line Integrals

Integrals over curves should not be confused with Vector Line Integrals. At the very least, the former yields a vector, while the latter yields a number. So, remember that the distinction between “integral over a curve” and “line integral” is crucial.

Integrals over Surfaces

TODO

Integrals over Volumes

TODO

Higher-Dimensional Integrals

TODO

Bibliography