Line, Surface and Volume Integral

QUIZ

1. Line Integrals

A line integral measures the value of a scalar or vector field along a curve.
The curve may be in 2D or 3D and is usually parametrized as $ \mathbf{r}(t) $.

For a scalar field $ \phi(x, y, z) $, the line integral along a curve $ C $ is:

\[\int_C \phi \, ds\]

where

\[ds = |\mathbf{r}'(t)|\, dt\]

This gives the accumulated value of the scalar field along the path.

For a vector field $ \mathbf{A} $, line integral is:

\[\int_C \mathbf{A} \cdot d\mathbf{r}\]

where $ d\mathbf{r} = dx\, \hat{i} + dy\, \hat{j} + dz\, \hat{k} $.

Interpretation:

Measures the work done by a force field along a path.
$W = \int_C \mathbf{F} \cdot d\mathbf{r}$
Circulation of a vector field.

Important: For a conservative field $ \mathbf{A} = \nabla \phi $:

\[\int_C \mathbf{A} \cdot d\mathbf{r} = \phi(B) - \phi(A)\]

i.e., path-independent.

1.1 Differential Length Elements $d\mathbf{l}$

Coordinate System	Differential Length Vector
Cartesian $(x, y, z)$	$ d\mathbf{l} = dx\,\hat{i} + dy\,\hat{j} + dz\,\hat{k} $
Cylindrical $(r, \theta, z)$	$ d\mathbf{l} = dr\,\hat{r} + r\,d\theta\,\hat{\theta} + dz\,\hat{z} $
Spherical $(r, \theta, \phi)$	$ d\mathbf{l} = dr\,\hat{r} + r\,d\theta\,\hat{\theta} + r\sin\theta\, d\phi\,\hat{\phi} $

2. Surface Integrals

A surface integral measures how a scalar or vector field interacts with a surface.

For a scalar field $ \phi $, over surface $ S $:

\[\iint_S \phi \, dS\]

where $ dS = |\mathbf{r}_u \times \mathbf{r}_v| \, du\, dv $.

Physical meaning:

Total mass on a thin sheet.
Total heat radiated by a surface.

Flux of vector field $ \mathbf{A} $ across a surface:

\[\iint_S \mathbf{A} \cdot d\mathbf{S}\]

where

\[d\mathbf{S} = \hat{n}\, dS\]

This measures how much of the field passes through the surface.

Examples:

Electric flux:
$\Phi_E = \iint_S \mathbf{E} \cdot d\mathbf{S}$
Magnetic flux:
$\Phi_B = \iint_S \mathbf{B} \cdot d\mathbf{S}$

Flux is positive: when the field flows along outward normal.
Flux is negative: when it flows inward.

2.1 Differential Surface Elements $d\mathbf{S}$

Surface elements depend on which surface is being used (constant coordinate).

Cartesian Coordinates

Surface	Differential Area Vector
$x=\text{constant}$	$ d\mathbf{S} = dy\,dz\, \hat{i} $
$y=\text{constant}$	$ d\mathbf{S} = dx\,dz\, \hat{j} $
$z=\text{constant}$	$ d\mathbf{S} = dx\,dy\, \hat{k} $

Cylindrical Coordinates

Surface	Differential Area Vector
$r=\text{constant}$	$ d\mathbf{S} = (r\, d\theta\, dz)\, \hat{r} $
$\theta=\text{constant}$	$ d\mathbf{S} = (dr\, dz)\, \hat{\theta} $
$z=\text{constant}$	$ d\mathbf{S} = (r\, dr\, d\theta)\, \hat{z} $

Spherical Coordinates

Surface	Differential Area Vector
$r=\text{constant}$	$ d\mathbf{S} = (r^2 \sin\theta\, d\theta\, d\phi)\, \hat{r} $
$\theta=\text{constant}$	$ d\mathbf{S} = (r \sin\theta\, dr\, d\phi)\, \hat{\theta} $
$\phi=\text{constant}$	$ d\mathbf{S} = (r\, dr\, d\theta)\, \hat{\phi} $

3. Volume Integrals

A volume integral gives the accumulated value of a field inside a 3D region.

\[\iiint_V \phi \, dV\]

For Cartesian:
$dV = dx\, dy\, dz$