Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. F = 3yi + yj+zk C: counterclockwise path around the boundary of the ellipse x2/4+y2/64=1

= 12π

Step-by-step explanation:

The field F is given by:

     (1)

The curve C is the ellipse:

In order to calculate the circulation of F around the curve C, you first find the parametric equation for the given ellipse.

The general form of an ellipse equation is:

The parametric equation is:

     (2)

The Stokes's theorem is given by the following identity:

The path integral is also:

     (3)

For F(r(t)) and dr(t) you obtain:

Next, in the equation (3) you obtain:

The circulation of the field around C is 12π

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation: