Line & Surface Integral Notation

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Parameterized Curves & Line Integrals

Curve

$\overrightarrow{r}\left(t\right)=⟨x\left(t\right),y\left(t\right),z\left(t\right)⟩$

Tangent Vector

$\overrightarrow{v}=\frac{\mathrm{d}\overrightarrow{r}}{\mathrm{d}t}=⟨\frac{\mathrm{d}x}{\mathrm{d}t},\frac{\mathrm{d}y}{\mathrm{d}t},\frac{\mathrm{d}z}{\mathrm{d}t}⟩$

Tangent Differential Vector

$\begin{array}{rl}\mathrm{d}\overrightarrow{s}=\mathrm{d}\overrightarrow{r}& =⟨\mathrm{d}x,\mathrm{d}y,\mathrm{d}z⟩\\ & =⟨\frac{\mathrm{d}x}{\mathrm{d}t},\frac{\mathrm{d}y}{\mathrm{d}t},\frac{\mathrm{d}z}{\mathrm{d}t}⟩\mathrm{d}t\\ & =\overrightarrow{v}\mathrm{d}t=\hat{v}\Vert \overrightarrow{v}\Vert \mathrm{d}t=\hat{v}\mathrm{d}s\end{array}$

Tangent Differential Scalar

Useful for arc length

$\begin{array}{rl}\mathrm{d}s=\Vert \mathrm{d}\overrightarrow{s}\Vert & =\sqrt{{\left(\mathrm{d}x\right)}^2+{\left(\mathrm{d}y\right)}^2+{\left(\mathrm{d}z\right)}^2}\\ & =\sqrt{{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)}^2+{\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)}^2+{\left(\frac{\mathrm{d}z}{\mathrm{d}t}\right)}^2}\mathrm{d}t\\ & =\Vert \overrightarrow{v}\Vert \mathrm{d}t\end{array}$

Arc Length Integral

$L=\underset{A}{\overset{B}{\int }}\mathrm{d}s=\underset{a}{\overset{b}{\int }}\Vert \overrightarrow{v}\Vert \mathrm{d}t$

Scalar Curve Integral

Integral of a scalar function $f(x,y,z)$ along $\overrightarrow{r}\left(t\right)$ from $A=\overrightarrow{r}\left(a\right)$ to $B=\overrightarrow{r}\left(b\right)$:

$\underset{A}{\overset{B}{\int }}f\mathrm{d}s=\underset{a}{\overset{b}{\int }}f\left(\overrightarrow{r}\left(t\right)\right)\Vert \overrightarrow{v}\Vert \mathrm{d}t$

Average Value

Average value of a function $f(x,y,z)$ along $\overrightarrow{r}\left(t\right)$ from $A=\overrightarrow{r}\left(a\right)$ to $B=\overrightarrow{r}\left(b\right)$:

$f_{\mathrm{a}\mathrm{v}\mathrm{e}}=\frac{1}{L}\underset{A}{\overset{B}{\int }}f\mathrm{d}s=\frac{1}{L}\underset{a}{\overset{b}{\int }}f\left(\overrightarrow{r}\left(t\right)\right)\Vert \overrightarrow{v}\Vert \mathrm{d}t$

Mass and Center of Mass

Mass density function $\rho$:

$\begin{array}{rl}M& =\underset{A}{\overset{B}{\int }}\rho \mathrm{d}s=\underset{a}{\overset{b}{\int }}\rho \Vert \overrightarrow{v}\Vert \mathrm{d}t\\ ⟨\overline{x},\overline{y},\overline{z}⟩& =\frac{1}{M}\underset{A}{\overset{B}{\int }}⟨x,y,z⟩\rho \mathrm{d}s\end{array}$

Special for 2D Curves

Normal Vector

Note: $\Vert \overrightarrow{n}\Vert =\Vert \overrightarrow{v}\Vert$

$\begin{array}{rl}\overrightarrow{n}={\overrightarrow{v}}^{\perp }& =\left|\begin{array}{cc}\hat{ı}& \hat{ȷ}\\ v_1& v_2\end{array}\right|\\ & =v_2\hat{ı}-v_1\hat{ȷ}& =⟨v_2,-v_1⟩\\ & =\frac{\mathrm{d}y}{\mathrm{d}t}\hat{ı}-\frac{\mathrm{d}x}{\mathrm{d}t}\hat{ȷ}& =⟨\frac{\mathrm{d}y}{\mathrm{d}t},-\frac{\mathrm{d}x}{\mathrm{d}t}⟩\end{array}$

Normal Differential Vector

$\begin{array}{rll}\mathrm{d}\overrightarrow{n}& =\mathrm{d}y\hat{ı}-\mathrm{d}x\hat{ȷ}& =⟨\mathrm{d}y,-\mathrm{d}x⟩\\ & =(\frac{\mathrm{d}y}{\mathrm{d}t}\hat{ı}-\frac{\mathrm{d}x}{\mathrm{d}t}\hat{ȷ})\mathrm{d}t& =⟨\frac{\mathrm{d}y}{\mathrm{d}t},-\frac{\mathrm{d}x}{\mathrm{d}t}⟩\mathrm{d}t\\ & =\overrightarrow{n}\mathrm{d}t=\hat{n}\Vert \overrightarrow{n}\Vert \mathrm{d}t=\hat{n}\mathrm{d}s\end{array}$

Normal Differential Scalar

$\mathrm{d}n=\Vert \mathrm{d}\overrightarrow{n}\Vert =\sqrt{{\left(\mathrm{d}y\right)}^2+{\left(\mathrm{d}x\right)}^2}=\mathrm{d}s$

Line Integral of Normal

Integral of the normal component of a vector field $\overrightarrow{G}=G_1\hat{ı}+G_2\hat{ȷ}=⟨G_1,G_2⟩$ along $\overrightarrow{r}\left(t\right)$:

$\begin{array}{rl}\underset{A}{\overset{B}{\int }}\overrightarrow{G}\cdot \mathrm{d}\overrightarrow{n}& =\underset{A}{\overset{B}{\int }}(G_1\mathrm{d}y-G_2\mathrm{d}x)\\ & =\underset{a}{\overset{b}{\int }}(G_1\frac{\mathrm{d}y}{\mathrm{d}t}-G_2\frac{\mathrm{d}x}{\mathrm{d}t})\mathrm{d}t\\ & =\underset{a}{\overset{b}{\int }}\overrightarrow{G}\cdot \overrightarrow{n}\mathrm{d}t\\ & =\underset{A}{\overset{B}{\int }}\overrightarrow{g}\cdot \hat{n}\mathrm{d}s\end{array}$

Furthermore, if $\overrightarrow{G}={\overrightarrow{F}}^{\perp }=F_2\hat{ı}-F_1\hat{ȷ}=⟨F_2,-F_1⟩$, then

$\overrightarrow{G}\cdot \overrightarrow{n}=⟨F_2,-F_1⟩\cdot ⟨v_2,-v_1⟩=\overrightarrow{F}\cdot \overrightarrow{v}$

and

$\underset{A}{\overset{B}{\int }}\overrightarrow{G}\cdot \mathrm{d}\overrightarrow{n}=\underset{A}{\overset{B}{\int }}(F_2\mathrm{d}y-(-F_1)\mathrm{d}x)=\underset{A}{\overset{B}{\int }}\overrightarrow{F}\cdot \mathrm{d}\overrightarrow{s}$

Parameterized Surfaces & Surface Integrals

Surface

$\overrightarrow{R}(s,t)=⟨x(s,t),y(s,t),z(s,t)⟩$

Tangent Vectors

$\begin{array}{rl}{\overrightarrow{e}}_s=\frac{\partial \overrightarrow{R}}{\partial s}& =⟨\frac{\partial x}{\partial s},\frac{\partial y}{\partial s},\frac{\partial z}{\partial s}⟩\\ {\overrightarrow{e}}_t=\frac{\partial \overrightarrow{R}}{\partial t}& =⟨\frac{\partial x}{\partial t},\frac{\partial y}{\partial t},\frac{\partial z}{\partial t}⟩\\ \end{array}$

Normal Vector

$\begin{array}{rl}\overrightarrow{N}={\overrightarrow{e}}_s\times {\overrightarrow{e}}_t& =\left|\begin{array}{ccc}\hat{ı}& \hat{ȷ}& \hat{k}\\ \frac{\partial x}{\partial s}& \frac{\partial y}{\partial s}& \frac{\partial z}{\partial s}\\ \frac{\partial x}{\partial t}& \frac{\partial y}{\partial t}& \frac{\partial z}{\partial t}\end{array}\right|\\ & =\frac{\partial (y,z)}{\partial (s,t)}\hat{ı}+\frac{\partial (z,x)}{\partial (s,t)}\hat{ȷ}+\frac{\partial (x,y)}{\partial (s,t)}\hat{k}\\ & =⟨\frac{\partial (y,z)}{\partial (s,t)},\frac{\partial (z,x)}{\partial (s,t)},\frac{\partial (x,y)}{\partial (s,t)}⟩\end{array}$

Surface Differential Vector

$\begin{array}{rl}\mathrm{d}\overrightarrow{S}& =⟨\mathrm{d}y\mathrm{d}z,\mathrm{d}z\mathrm{d}x,\mathrm{d}x\mathrm{d}y⟩=⟨\frac{\partial (y,z)}{\partial (s,t)},\frac{\partial (z,x)}{\partial (s,t)},\frac{\partial (x,y)}{\partial (s,t)}⟩\mathrm{d}s\mathrm{d}t\\ & =\overrightarrow{N}\mathrm{d}s\mathrm{d}t=\hat{N}\Vert \overrightarrow{N}\Vert \mathrm{d}s\mathrm{d}t=\hat{N}\mathrm{d}S\end{array}$

Surface Differential Scalar

$\begin{array}{rl}\mathrm{d}S=\Vert \mathrm{d}\overrightarrow{S}\Vert & =\sqrt{{\left(\mathrm{d}y\mathrm{d}z\right)}^2+{\left(\mathrm{d}z\mathrm{d}x\right)}^2+{\left(\mathrm{d}x\mathrm{d}y\right)}^2}\\ & =\sqrt{{\left(\frac{\partial (y,z)}{\partial (s,t)}\right)}^2+{\left(\frac{\partial (z,x)}{\partial (s,t)}\right)}^2+{\left(\frac{\partial (x,y)}{\partial (s,t)}\right)}^2}\\ & =\Vert \overrightarrow{N}\Vert \mathrm{d}s\mathrm{d}t\end{array}$

Surface Area Integral

$A=\underset{\overrightarrow{R}}{\iint }\mathrm{d}S=\underset{\overrightarrow{R}}{\iint }\Vert \overrightarrow{N}\Vert \mathrm{d}s\mathrm{d}t$

Scalar Surface Integral

Integral of a scalar function $f(x,y,z)$ over $\overrightarrow{R}(s,t)$:

$\underset{\overrightarrow{R}}{\iint }f\mathrm{d}S=\underset{\overrightarrow{R}}{\iint }f\left(\overrightarrow{R}(s,t)\right)\Vert \overrightarrow{N}\Vert \mathrm{d}s\mathrm{d}t$

Average Value

Average value of a function $f(x,y,z)$ over $\overrightarrow{R}(s,t)$:

$f_{\mathrm{a}\mathrm{v}\mathrm{e}}=\frac{1}{A}\underset{\overrightarrow{R}}{\iint }f\mathrm{d}S=\frac{1}{A}\underset{\overrightarrow{R}}{\iint }f\left(\overrightarrow{R}(s,t)\right)\Vert \overrightarrow{n}\Vert \mathrm{d}s\mathrm{d}t$

Mass and Center of Mass

Mass density function $\rho$:

$\begin{array}{rl}M& =\underset{\overrightarrow{R}}{\iint }\rho \mathrm{d}S=\underset{\overrightarrow{R}}{\iint }\rho \Vert \overrightarrow{N}\Vert \mathrm{d}s\mathrm{d}t\\ ⟨\overline{x},\overline{y},\overline{z}⟩& =\frac{1}{M}\underset{\overrightarrow{R}}{\iint }⟨x,y,z⟩\rho \mathrm{d}S\end{array}$

Vector Surface Integral (Flux)

Integral of a vector field $\overrightarrow{F}=⟨F_1,F_2,F_3⟩$ over $\overrightarrow{R}(u,v)$:

$\begin{array}{rl}\underset{\overrightarrow{R}}{\iint }\overrightarrow{F}\cdot \mathrm{d}\overrightarrow{S}& =\underset{\overrightarrow{R}}{\iint }(F_1\mathrm{d}y\mathrm{d}z+F_2\mathrm{d}z\mathrm{d}x+F_3\mathrm{d}x\mathrm{d}y)\\ & =\underset{\overrightarrow{R}}{\iint }(F_1\frac{\partial (y,z)}{\partial (s,t)}+F_2\frac{\partial (z,x)}{\partial (s,t)}+F_3\frac{\partial (x,y)}{\partial (s,t)})\mathrm{d}s\mathrm{d}t\\ & =\underset{\overrightarrow{R}}{\iint }\overrightarrow{F}\cdot \overrightarrow{N}\mathrm{d}s\mathrm{d}t\\ & =\underset{\overrightarrow{R}}{\iint }\overrightarrow{F}\cdot \hat{N}\mathrm{d}S\end{array}$