Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = (x + y2)i + (y + z2)j + (z + x2)k, C is the triangle with vertices (5, 0, 0), (0, 5, 0), and (0, 0, 5).

Step-by-step explanation:

Given that:

  , where C is the triangle with vertices (5, 0, 0), (0, 5, 0), and (0, 0, 5).

The objective is to use Stokes' Theorem  to evaluate CF. dr

Stokes Theorem :

To estimate curl F , we need to find the partial derivatives:

So;

partial derivative is:

partial derivative is:

partial derivative is:

These resulted into

curl F = (0 - 2z)i + ( 0 -2x) j + ( 0 - 2y) k

= ( -2z, -2x, -2y )

The normal  vector and the equation of the plane can be expressed as follows:

If a = (0,5,0 - ( 5,0,0)

a = ( -5,5,0 )

Also ,

b = (0, 0,5) - (5,0,0)

b = (-5. 0,5)

However,

a × b = (25 - 0)i - (-25-0)j+ (0+25)k

a × b = 25i +25j +25k

∴ the normal vector can be n = (1,1,1)

If we assume x to be x = (x,y,z)

and

Then

collecting like terms

x +y +z  = 5

now, it is vivid that from the equation , the plane of the  normal vector =(1,1,1)

Similarly, x+y+z = 5 is the projection of surface on the xy - plane such that the line x +y = 5

Thus; the domain D = {(x,y) | 0 ≤ x ≤ 5, 0 ≤ y ≤ 5 - x}

To evaluate the line integral using Stokes' Theorem

g = (2, 2)

step-by-step explanation:

given the ratio is 1 : 1 then f is the midpoint of eg

let the coordinates of g = (x, y)

using the midpoint formula

(0 + x) = 1 ( multiply both sides by 2 )

0 + x = 2 ⇒ x = 2

similarly

(4 + y) = 3 ( multiply both sides by 2 )

4 + y = 6 ( subtract 4 from both sides )

y = 2

coordinates of g = (2, 2)