| 422 |
path integral
|
\[
\int_{\vect{c}}f\left(x,y,z\right)\ds
=\int_a^b f(\vect{c}(t))\norm{\vect{c}'(t)}\dt
\]
|
| 424 |
area of a fence
|
\[
\int_{\vect{c}}f\left(x,y\right)\ds
=\int_a^b f(\vect{c}(t))
\norm{\vect{c}'(t)}\dt
\]
|
| 431 |
line integral.
$\vect{T}\left(t\right)$
$=\vect{c}'t/\norm{\vect{c}'(t)}$
is the unit tangent vector to $c.$
|
\[
\begin{multline*}
\int_{\vect{c}}{\vect{F}\cdot\,d\vect{s}}
=\int_{\vect{c}}
{\vect{F}\cdot\vect{T}\ds}\\
=\int_a^b
\vect{F}\left(\vect{c}\left(t\right)\right)
\cdot\vect{c}'\left(t\right)
\dt\\
=\int_a^b\vect{F}(\vect{c}(t))
\cdot\vect{T}(t)
\norm{c'(t)}
\dt
\end{multline*}
\]
|
| 430 |
work done by force field $F$
|
\[
W=\int_{\vect{c}}
\vect{F}\cdot\,d\vect{s}
=\int_a^b
\vect{F}\left(\vect{c}\left(t\right)\right)
\cdot\vect{c}'\left(t\right)
\dt
\]
|
| 432 |
integral of a differential form along a path
$c$
|
\[
\begin{multline*}
\int_{\vect{c}}{\vect{F}\cdot d\vect{s}}
=\int_{\vect{c}}
F_1\dx+F_2\dy+F_3\dz\\
=\int_a^b
\left(
F_1\frac{dx}{dt}
+F_2\frac{dy}{dt}
+F_3\frac{dz}{dt}
\right)
\dt
\end{multline*}
\]
|
| 437 |
reparametrization definition
where
$h:I\rightarrow I_1 C1$ is one-to-one and onto
and
$\vect{c}:I_1\rightarrow\R^3$
is a piecewise $C^1$ map
|
\[
\vect{p}=\vect{c}\circ h:I\rightarrow\R^3
\]
|
| 435 |
orientation preserving reparametrization
where
$\vect{c}:\left[a_1,b_1\right]\rightarrow\R^3$
and
$\vect{p}:\left[a,b\right]\rightarrow\R^3$
|
\[
\vect{p}\left(a\right)=\vect{c}\left(a_1\right)\\
\text{and}\\
\vect{p}\left(b\right)=\vect{c}\left(b_1\right)
\]
|
| 435 |
orientation reversing reparametrization
where
$\vect{c}:\left[a_1,b_1\right]\rightarrow\R^3$
and
$\vect{p}:\left[a,b\right]\rightarrow\R^3$
|
\[
\vect{p}\left(a\right)=\vect{c}\left(b_1\right)\\
\text{and}\\
\vect{p}\left(b\right)=\vect{c}\left(a_1\right)
\]
|
| 437 |
line integral for orientation preserving change
of parametrization,
where $F$ is a vector field continuous on the $C^1$ path
$\vect{c}:\left[a_1,b_1\right]\rightarrow\R^3$
and
$\vect{p}:\left[a,b\right]\rightarrow\R^3$
is an orientation preserving reparametrization of $c.$
|
\[
\int_{\vect{p}}{\vect{F}\cdot d\vect{s}}
=\int_{\vect{c}}{\vect{F}\cdot d\vect{s}}
\]
|
| 437 |
line integral for orientation reversing change of
parametrization, where $F$ is a vector field
continuous on the $C^1$ path
$\vect{c}:\left[a_1,b_1\right]\rightarrow\R^3$
and
$\vect{p}:\left[a,b\right]\rightarrow\R^3$
is an orientation reversing reparametrization
of $c.$
|
\[
\int_{\vect{p}}{\vect{F}\cdot d\vect{s}}
=-\int_{\vect{c}}{\vect{F}\cdot d\vect{s}}
\]
|
| 439 |
path integral for reparametrization.
This says that path integrals don't change under
any reparametrization.
|
\[
\int_{\vect{c}}f\left(x,y,z\right)\ds
=\int_{\vect{p}}f\left(x,y,z\right)\ds
\]
|
| 440 |
line integrals of gradient vector fields.
Recall that a vector field $F$ is a gradient vector
field if
$\vect{F}=\mathrm{\nabla f}$
for some real-valued function
$f$.)
|
\[
\begin{multline*}
\int_\vect{c}
\nabla f\cdot d\vect{s}
=\int_a^b
\nabla f
\left(\vect{c}\left(t\right)\right)
\vect{c}'
\left(t\right)
\dt\\
=f\left(\vect{c}\left(b\right)\right)
-f\left(\vect{c}\left(a\right)\right)
\end{multline*}
\]
|
| 442 |
line and path integrals over oriented
simple and simple closed curves,
where $C$ is a simple or
simple closed curve parametrized by the orientation
preserving $c.$ This says that the integral doesn't
change for any orientation preserving
parametrization $c.$
|
\[
\int_{C}{\vect{F}\cdot d\vect{s}}
=\int_{\vect{c}}
\vect{F}\cdot d\vect{s}\\
\text{and}\\
\int_{C}f\ds=\int
_{\vect{c}}f\ds
\]
|
| 444 |
line integrals over curves with opposite orientations
|
\[
\int_{C}{\vect{F}\cdot d\vect{s}}
=-\int_{C-}{\vect{F}\cdot d\vect{s}}
\]
|
| 444 |
line integral over piecewise simple curves
|
\[
\begin{multline*}
\int_{C}{\vect{F}\cdot d\vect{s}}
=\sum_{n=1}^{k}\int_{C_n}
\vect{F}\cdot d\vect{s}\\
=\int_{C_1}{\vect{F}\cdot d\vect{s}}
+\int_{C_2}{\vect{F}\cdot d\vect{s}}
+\cdots+\int_{C_k}{\vect{F}\cdot d\vect{s}}
\end{multline*}
\]
|
| 453 |
parametrized surface
|
\[
S=\mathrm{\Phi}\left(D\right)
\]
|
| 454 |
parametric equation of a plane
|
\[
\mathrm{\Phi}\left(u,v\right)=u\vect{a}+v\vect{b}+\vect{c}
\]
|
| 454 |
equation of a plane (standard form)
derived from a parametrization of the plane
|
\[
\vect{N}\cdot\left(x-x_0,y-y_0,z-z_0\right)=0\\
A\left(x-x_0\right)+B\left(y-y_0\right)+C\left(z-z_0\right)=0\\
\text{where}\\
\vect{N}
=\vect{a}\times\vect{b}
=A\vect{i}
+B\vect{j}
+C\vect{k}\\
\text{and}\\
\Phi\left(u,v\right)
=u\vect{a}
+v\vect{b}
+\vect{c}
\]
|
| 454 |
tangent vector to surface
at $\Phi\left(u_0,v_0\right)$
and tangent to curve at
$\Phi\left(u_0,v_0\right)$
and to the curve
$\vect{\Phi}\left(\vect{u}_\zeros,\vect{t}\right)$
contained by the surface.
|
\[
\begin{multline*}
\vect{T}_v=\frac{\partial\Phi}{\partial v}
=\frac{\partial x}{\partial v}\left(u_0,v_0\right)
\vect{i}\\
+\frac{\partial y}{\partial v}\left(u_0,v_0\right)
\vect{j}
+\frac{\partial z}{\partial v}\left(u_0,v_0\right)
\vect{k}
\end{multline*}
\]
|
| 454 |
tangent vector to surface
at $\mathrm{\Phi}\left(u_0,v_0\right)$
and to the curve
$\vect{\Phi}\left(\vect{t},\vect{v}_\zeros\right)$
contained by the surface.
|
\[
\begin{multline*}
\vect{T}_u
=\frac{\partial\Phi}{\partial u}
=\frac{\partial x}{\partial u}\left(u_0,v_0\right)
\vect{i}\\
+\frac{\partial y}{\partial u}\left(u_0,v_0\right)
\vect{j}
+\frac{\partial z}{\partial u}\left(u_0,v_0\right)
\vect{k}
\end{multline*}
\]
|
| 454 |
|
\[
\vect{N}\cdot\left(x-x_0,y-y_0,z-z_0\right)=0\\
A\left(x-x_0\right)+B\left(y-y_0\right)+C\left(z-z_0\right)=0\\
\text{where}\\
\vect{N}=\vect{T}_u\times\vect{T}_v
=A\vect{i}+B\vect{j}+C\vect{k}\\
\text{and}\\
\Phi\left(u,v\right)
=u\vect{a}+v\vect{b}+\vect{c}\\
\]
|
| 457 |
parametrization of the graph of a function
$g:\R^2\rightarrow\R$
|
\[
x=u,y=v,z=g\left(u,v\right)
\]
|
| 461 |
area of a surface over a region
$D$
|
\[
A\left(S\right)
=\iint_D\norm{\vect{T}_u\times\vect{T}_v}
\du\dv
\]
|
| 461 |
area of a surface over a piecewise defined region
|
\[
A\left(S\right)=\sum A\left(S_i\right)
\]
|
| 462 |
the norm
$\norm{\vect{T}_u\times\vect{T}_v}$
|
\[
\norm{\vect{T}_u\times\vect{T}_v}
=\sqrt{
\left[
\frac
{\partial(x,y)}
{\partial(u,v)}
\right]^2
+ \left[
\frac
{\partial(y,z)}
{\partial(u,v)}
\right]^2
+ \left[
\frac
{\partial(x,z)}
{\partial(u,v)}
\right]^2
}
\]
|
| 465 |
surface area of a graph
where $x=u,$ $y=v,$ and
$z=g\left(u,v\right)$
|
\[
A\left(S\right)
=\iint_{D}
\left(
\sqrt{
\left(
\frac{\partial g}{\partial x}
\right)^2
+\left(
\frac{\partial g}{\partial y}
\right)^2
+1
}
\right)
\,dA
\]
|
| 467 |
surface of revolution
|
\[
\begin{multline*}
A(S)=2\pi\int_a^b
\abs{f\left(x\right)}
\sqrt{1+\left[f'\left(x\right)\right]^2}
\dx\\
=\int_{\vect{c}}
2\pi
\abs{f\left(x\right)}
\ds
\end{multline*}
\]
|
| 474 |
integral of scalar function over a surface
|
\[
\begin{multline*}
\iint_S{f\left(x,y,z\right)\,dS}\\
=\iint_D
f\left(\Phi(u,v)\right)
\norm{\vect{T}_u\times\vect{T}_v}
\du\dv
\end{multline*}
\]
|
| 476 |
integrals over graphs
|
\[
\]
|
| 484 |
surface integral of vector fields
|
\[
\iint_{\mathrm{\Phi}}{\vect{F}\cdot d\vect{S}}
=\iint_{D}
\vect{F}\cdot
\left(
\vect{T}_u\times\vect{T}_v
\right)
\du\dv
\]
|
Alex Notes