Verify that Stokes’ Theorem is true for the given vector field F and surface S.
13. F(x, y, z) = -y i + x j -2 k,
S is the cone z2 = x2 + y2, 0 ≤ z ≤ 4, oriented downward
To verify: Whether Stokes’ theorem is true for given vector filed and surface S.
The field is and
Consider the expression for surface of cone, is,
Write the expression for curl of .
Write the expression for the Stokes’ theorem.
S is surface.
Consider surface S, is in upward orientation. Write the expression for surface integral of over surface S.
A is area.
The boundary value of surface cone is a circle. The maximum value of z is 4.
Substitute 4 for z in equation (1),
Consider the parametric equations since the orientation of surface is downward as,
Write the expression for vector function .
Substitute for x, for y, 4 for z,
Differentiate the expression with respect to t,
Find the value of .
Find the value of .
Write the expression for
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