Line integral

Recall: Riemann integral

Notation:

Def: Line integral: We denote path $C$ joining A and B into N small line element $\Delta \bold r_p = 1, 2, \dots N$, If a field $\bold a(x_p, y_p, z_p)$ is any point on the line element $\Delta r_p$, then

Remark: If $C$ is closed, then

where $C$ is anticlockwise

Properties:

3. (could be) path independent

How to evaluate:

Example:

evaluate $I=\int_C \bold a \cdot d\bold r$, where $\bold a = \left[\begin{array}&x+y\\y-x\end{array}\right]$, along the curve $\left\{\begin{array}& x=2u^2+u+1\\ y=1+u^2 \end{array}\right.$ from $(1,1)$ to $(4.2)$.

Physics example:

Line integrals with respect to a scalar

Notion:

where $ds$ is arc length along the curve $C$

How to solve:

where $u$ is the parameter of $C$ that can be represented by $C: \bold r (u)$

Example:

evaluate $I = \int_C (x-y)^2 ds$, where $C$ is the semicircle from $A(a,0)$ to $B(-a,0)$ for $y\geq 0$.

Green theorem in a plane

Green theorem:

where

Conservative fields and potentials

Def: Region $R$ is conservative if and only if

1. $\int_A^B \bold a \cdot d \bold r$ is path independent $\Rightarrow$ $\oint_C \bold a \cdot d\bold r = 0$
2. $\bold a = \nabla \phi$ where $\phi$ is a single valued function
3. $\nabla \times \bold a=0$
4. $\bold a \cdot d\bold r$ is exact differential

Example:

Find $\phi$ if $\bold a = \nabla \phi = (xy^2+z)\bold i + (x^2y+2)\bold y + x\bold k$

$\phi$ must satisfy

Surface integrals

Notion:

where the direction of $\bold S$ is parpendicular to the surface

Def:

How to evaluate:

find $I= \int_S \bold a \cdot d\bold S$, where $\bold a = x\bold i$, $S: x^2+y^2+z^2=a^2, z\leq 0$.

Using spherical coordinate

Also, more generally,

convert to polar coordinate:

where $\rho = a\sin u$

Vector areas of surfaces

Def:

Remark: for a closed surface, $\bold S =0$

$\bold S$ is only depend on its boundary curve:

Volume integrals

Notion:

Volumes of three-dimensional regions

Integral forms for grad., div. and curl

where V is a small volume enclosing P and S is its bounding surface.

it can be shown in Cartisian, Cylindrical and Spherical Coordinates.

Divergence theorem and related theorem

1. $\bold a$ is a vector field which is continous and diffentiable in $V \subset \mathbb{R^3}$

2. $V$ is divided into a large number of small volumes $V_i$

3. By defination of div.

   $S_i$ is surface of $V_i$

4. By summing over $i$, we have

5. Divergence theorem holds for simply and multiply connected regions

example:

evaluate $I=\int_S \bold a \cdot d\bold S$, where $\bold a = (y-x, xz, z-x^2)$ and $S$ is a open surface $x^2+y^2+z^2=z^2, z\leq 0$.

 

 

Green theorems

Other related integral transforms

Stokes' theorem and related theorems

related integral theorems

Last Updated on 3 years by Yichen Liu