Line & Surface Integral Notation

Download PDF: Local | Remote

This page is ©2001-10 P. Yasskin and is available on his web page.

Parameterized Curves & Line Integrals

Curve

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{r}\left(t\right) = \left\langle x\left(t\right), y\left(t\right), z\left(t\right) \right\rangle}

Tangent Vector

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}

Tangent Differential Scalar

Useful for arc length

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{G} = G_1\,\hat\imath + G_2 \,\hat\jmath = \left\langle G_1, G_2 \right\rangle} along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{r}\left(t\right)} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}

Furthermore, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{G} = \vec{F}^\perp = F_2 \,\hat\imath - F_1 \,\hat\jmath = \left\langle F_2, -F_1 \right\rangle} , then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}

Normal Vector

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}

Surface Differential Vector

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}

Surface Differential Scalar

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}

Surface Area Integral

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(x,y,z\right)} over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{R}\left(s,t\right)} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(x,y,z\right)} over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{R}\left(s,t\right)} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align}}

Vector Surface Integral (Flux)

Integral of a vector field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{F} = \left\langle F_1, F_2, F_3 \right\rangle} over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{R}\left(u,v\right)} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}