Line Integrals

First type:  Scalar --> scalar.   It is independent of the direction of the curve.

Second type : Vector --> scalar. When we change the direction of the curve, the result differ by a sign.

 

Parametric form of curves in 2D and 3D

Line segment from (x1, y1,z1) to (x2,y2,z2)

    x = x1 + (x2-x1) t
    y = y1 + (y2-y1) t,                    0 <= t <= 1.
    z = z1 + (z2-z1) t
 

     y = f(x)   or x = f(y)

    x= x                                               or           x = f(y)
    y = f(x)                                                         y = y
 

circles, ellipses, special closed curves, helix ...

differentials:  arc-length,    ds = |r'(t)| dt;     dr =<dx, dy, dz>;   dx = x'(t) dt, ....

 

Several pieces, break-up, if closed, consider to use the Green's theorem

Magic vector operator:

Multiply to a function: Gradient,             grad f

Dot product:  Divergence,   div F = 0,  F  is divergence free/incompressible.    Curl F is always divergence free.

Cross product: Curl F,       Curl F = 0,   F is irrotational/conservative.

 

If F is conservative,  ( curl F = 0,  P_y = Q_x in 2D) then:

Line integral is independent of path. We can choose a special path, for example, x=a, then y=b.

Line integral is zero for closed curves.

There is a potential function whose gradient is F.

Find a potential function and use it for line integrals

Step 1: Check whether F is conservative or not.

Step 2: Set  f_x = P,  integrate with x to get

             f =    ?     +   c

Step 3: Differentiate above with y and equate it to Q.

Step 4: Solve for  c_y

Step 5: Plug c back into f.

    .......

Step  6: Check

Surface Integrals:

Parametric form of surfaces:

z = f(x,y)

f(x,y,z) = 0

Special surfaces and their normal directions: Spheres, Cylinders, ellipsoids etc.

Tangent plane and differential area.

Total area of a surface.