Line & Surface Integral Notation

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Parameterized Curves & Line Integrals

Curve

${\vec {r}}\left(t\right)=\left\langle x\left(t\right),y\left(t\right),z\left(t\right)\right\rangle$

Tangent Vector

${\vec {v}}={\frac {\mathrm {d} {\vec {r}}}{\mathrm {d} t}}=\left\langle {\frac {\mathrm {d} x}{\mathrm {d} t}},{\frac {\mathrm {d} y}{\mathrm {d} t}},{\frac {\mathrm {d} z}{\mathrm {d} t}}\right\rangle$

Tangent Differential Vector

${\begin{aligned}\mathrm {d} {\vec {s}}=\mathrm {d} {\vec {r}}&=\left\langle \mathrm {d} x,\mathrm {d} y,\mathrm {d} z\right\rangle \\&=\left\langle {\frac {\mathrm {d} x}{\mathrm {d} t}},{\frac {\mathrm {d} y}{\mathrm {d} t}},{\frac {\mathrm {d} z}{\mathrm {d} t}}\right\rangle \mathrm {d} t\\&={\vec {v}}\,\mathrm {d} t={\hat {v}}\left\|{\vec {v}}\right\|\,\mathrm {d} t={\hat {v}}\,\mathrm {d} s\end{aligned}}$

Tangent Differential Scalar

Useful for arc length

${\begin{aligned}\mathrm {d} s=\left\|\mathrm {d} {\vec {s}}\right\|&={\sqrt {\left(\mathrm {d} x\right)^{2}+\left(\mathrm {d} y\right)^{2}+\left(\mathrm {d} z\right)^{2}}}\\&={\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} z}{\mathrm {d} t}}\right)^{2}}}\,\mathrm {d} t\\&=\left\|{\vec {v}}\right\|\,\mathrm {d} t\end{aligned}}$

Arc Length Integral

$L=\int _{A}^{B}\mathrm {d} s=\int _{a}^{b}\left\|{\vec {v}}\right\|\,\mathrm {d} t$

Scalar Curve Integral

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

$\int _{A}^{B}f\,\mathrm {d} s=\int _{a}^{b}f\left({\vec {r}}\left(t\right)\right)\left\|{\vec {v}}\right\|\,\mathrm {d} t$

Average Value

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

$f_{\mathrm {ave} }={\frac {1}{L}}\int _{A}^{B}f\,\mathrm {d} s={\frac {1}{L}}\int _{a}^{b}f\left({\vec {r}}\left(t\right)\right)\left\|{\vec {v}}\right\|\,\mathrm {d} t$

Mass and Center of Mass

Mass density function $\rho$:

${\begin{aligned}M&=\int _{A}^{B}\rho \,\mathrm {d} s=\int _{a}^{b}\rho \left\|{\vec {v}}\right\|\,\mathrm {d} t\\\left\langle {\bar {x}},{\bar {y}},{\bar {z}}\right\rangle &={\frac {1}{M}}\int _{A}^{B}\left\langle x,y,z\right\rangle \rho \,\mathrm {d} s\end{aligned}}$

Special for 2D Curves

Normal Vector

Note: $\left\|{\vec {n}}\right\|=\left\|{\vec {v}}\right\|$

${\begin{array}{rll}{\vec {n}}={\vec {v}}^{\perp }&={\begin{vmatrix}{\hat {\imath }}&{\hat {\jmath }}\\v_{1}&v_{2}\end{vmatrix}}\\[2pt]&=v_{2}\,{\hat {\imath }}-v_{1}\,{\hat {\jmath }}&=\left\langle v_{2},-v_{1}\right\rangle \\[2pt]&={\dfrac {\mathrm {d} y}{\mathrm {d} t}}\,{\hat {\imath }}-{\dfrac {\mathrm {d} x}{\mathrm {d} t}}\,{\hat {\jmath }}&=\left\langle {\dfrac {\mathrm {d} y}{\mathrm {d} t}},-{\dfrac {\mathrm {d} x}{\mathrm {d} t}}\right\rangle \end{array}}$

Normal Differential Vector

${\begin{array}{rll}\mathrm {d} {\vec {n}}&=\mathrm {d} y\,{\hat {\imath }}-\mathrm {d} x\,{\hat {\jmath }}&=\left\langle \mathrm {d} y,-\mathrm {d} x\right\rangle \\[2pt]&=\left({\dfrac {\mathrm {d} y}{\mathrm {d} t}}\,{\hat {\imath }}-{\dfrac {\mathrm {d} x}{\mathrm {d} t}}\,{\hat {\jmath }}\right)\,\mathrm {d} t&=\left\langle {\dfrac {\mathrm {d} y}{\mathrm {d} t}},-{\dfrac {\mathrm {d} x}{\mathrm {d} t}}\right\rangle \,\mathrm {d} t\\[4pt]&={\vec {n}}\,\mathrm {d} t={\hat {n}}\left\|{\vec {n}}\right\|\,\mathrm {d} t={\hat {n}}\,\mathrm {d} s\end{array}}$

Normal Differential Scalar

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

Line Integral of Normal

Integral of the normal component of a vector field ${\vec {G}}=G_{1}\,{\hat {\imath }}+G_{2}\,{\hat {\jmath }}=\left\langle G_{1},G_{2}\right\rangle$ along ${\vec {r}}\left(t\right)$:

${\begin{aligned}\int _{A}^{B}{\vec {G}}\cdot \mathrm {d} {\vec {n}}&=\int _{A}^{B}\left(G_{1}\,\mathrm {d} y-G_{2}\,\mathrm {d} x\right)\\&=\int _{a}^{b}\left(G_{1}\,{\frac {\mathrm {d} y}{\mathrm {d} t}}-G_{2}\,{\frac {\mathrm {d} x}{\mathrm {d} t}}\right)\,\mathrm {d} t\\&=\int _{a}^{b}{\vec {G}}\cdot {\vec {n}}\,\mathrm {d} t\\&=\int _{A}^{B}{\vec {g}}\cdot {\hat {n}}\,\mathrm {d} s\end{aligned}}$

Furthermore, if ${\vec {G}}={\vec {F}}^{\perp }=F_{2}\,{\hat {\imath }}-F_{1}\,{\hat {\jmath }}=\left\langle F_{2},-F_{1}\right\rangle$, then

${\vec {G}}\cdot {\vec {n}}=\left\langle F_{2},-F_{1}\right\rangle \cdot \left\langle v_{2},-v_{1}\right\rangle ={\vec {F}}\cdot {\vec {v}}$

and

$\int _{A}^{B}{\vec {G}}\cdot \mathrm {d} {\vec {n}}=\int _{A}^{B}\left(F_{2}\,\mathrm {d} y-\left(-F_{1}\right)\,\mathrm {d} x\right)=\int _{A}^{B}{\vec {F}}\cdot \mathrm {d} {\vec {s}}$

Parameterized Surfaces & Surface Integrals

Surface

${\vec {R}}\left(s,t\right)=\left\langle x\left(s,t\right),y\left(s,t\right),z\left(s,t\right)\right\rangle$

Tangent Vectors

${\begin{aligned}{\vec {e}}_{s}={\frac {\partial {\vec {R}}}{\partial s}}&=\left\langle {\frac {\partial x}{\partial s}},{\frac {\partial y}{\partial s}},{\frac {\partial z}{\partial s}}\right\rangle \\{\vec {e}}_{t}={\frac {\partial {\vec {R}}}{\partial t}}&=\left\langle {\frac {\partial x}{\partial t}},{\frac {\partial y}{\partial t}},{\frac {\partial z}{\partial t}}\right\rangle \\\end{aligned}}$

Normal Vector

${\begin{aligned}{\vec {N}}={\vec {e}}_{s}\times {\vec {e}}_{t}&={\begin{vmatrix}{\hat {\imath }}&{\hat {\jmath }}&{\hat {k}}\\{\frac {\partial x}{\partial s}}&{\frac {\partial y}{\partial s}}&{\frac {\partial z}{\partial s}}\\{\frac {\partial x}{\partial t}}&{\frac {\partial y}{\partial t}}&{\frac {\partial z}{\partial t}}\end{vmatrix}}\\&={\frac {\partial \left(y,z\right)}{\partial \left(s,t\right)}}\,{\hat {\imath }}+{\frac {\partial \left(z,x\right)}{\partial \left(s,t\right)}}\,{\hat {\jmath }}+{\frac {\partial \left(x,y\right)}{\partial \left(s,t\right)}}\,{\hat {k}}\\&=\left\langle {\frac {\partial \left(y,z\right)}{\partial \left(s,t\right)}},{\frac {\partial \left(z,x\right)}{\partial \left(s,t\right)}},{\frac {\partial \left(x,y\right)}{\partial \left(s,t\right)}}\right\rangle \end{aligned}}$

Surface Differential Vector

${\begin{aligned}\mathrm {d} {\vec {S}}&=\left\langle \mathrm {d} y\,\mathrm {d} z,\mathrm {d} z\,\mathrm {d} x,\mathrm {d} x\,\mathrm {d} y\right\rangle =\left\langle {\frac {\partial \left(y,z\right)}{\partial \left(s,t\right)}},{\frac {\partial \left(z,x\right)}{\partial \left(s,t\right)}},{\frac {\partial \left(x,y\right)}{\partial \left(s,t\right)}}\right\rangle \,\mathrm {d} s\,\mathrm {d} t\\&={\vec {N}}\,\mathrm {d} s\,\mathrm {d} t={\hat {N}}\left\|{\vec {N}}\right\|\,\mathrm {d} s\,\mathrm {d} t={\hat {N}}\,\mathrm {d} S\end{aligned}}$

Surface Differential Scalar

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

Surface Area Integral

$A=\iint _{\vec {R}}\mathrm {d} S=\iint _{\vec {R}}\left\|{\vec {N}}\right\|\,\mathrm {d} s\,\mathrm {d} t$

Scalar Surface Integral

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

$\iint _{\vec {R}}f\,\mathrm {d} S=\iint _{\vec {R}}f\left({\vec {R}}\left(s,t\right)\right)\left\|{\vec {N}}\right\|\,\mathrm {d} s\,\mathrm {d} t$

Average Value

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

$f_{\mathrm {ave} }={\frac {1}{A}}\iint _{\vec {R}}f\,\mathrm {d} S={\frac {1}{A}}\iint _{\vec {R}}f\left({\vec {R}}\left(s,t\right)\right)\left\|{\vec {n}}\right\|\,\mathrm {d} s\,\mathrm {d} t$

Mass and Center of Mass

Mass density function $\rho$:

${\begin{aligned}M&=\iint _{\vec {R}}\rho \,\mathrm {d} S=\iint _{\vec {R}}\rho \left\|{\vec {N}}\right\|\,\mathrm {d} s\,\mathrm {d} t\\\left\langle {\bar {x}},{\bar {y}},{\bar {z}}\right\rangle &={\frac {1}{M}}\iint _{\vec {R}}\left\langle x,y,z\right\rangle \rho \,\mathrm {d} S\end{aligned}}$

Vector Surface Integral (Flux)

Integral of a vector field ${\vec {F}}=\left\langle F_{1},F_{2},F_{3}\right\rangle$ over ${\vec {R}}\left(u,v\right)$:

${\begin{aligned}\iint _{\vec {R}}{\vec {F}}\cdot \mathrm {d} {\vec {S}}&=\iint _{\vec {R}}\left(F_{1}\,\mathrm {d} y\,\mathrm {d} z+F_{2}\,\mathrm {d} z\,\mathrm {d} x+F_{3}\,\mathrm {d} x\,\mathrm {d} y\right)\\&=\iint _{\vec {R}}\left(F_{1}\,{\frac {\partial \left(y,z\right)}{\partial \left(s,t\right)}}+F_{2}\,{\frac {\partial \left(z,x\right)}{\partial \left(s,t\right)}}+F_{3}\,{\frac {\partial \left(x,y\right)}{\partial \left(s,t\right)}}\right)\,\mathrm {d} s\,\mathrm {d} t\\&=\iint _{\vec {R}}{\vec {F}}\cdot {\vec {N}}\,\mathrm {d} s\,\mathrm {d} t\\&=\iint _{\vec {R}}{\vec {F}}\cdot {\hat {N}}\,\mathrm {d} S\end{aligned}}$