Verify that Stokes’ Theorem is true for the given vector field F and surface S.
14. F(x, y, z) = -2yz i + y j + 3x k,
S is the part of the paraboloid z = 5 - x2 - y2 that lies above the plane z = 1, oriented upward
To verify: Whether Stokes’ theorem is true for given vector filed and surface S.
The field is and
Consider the expression for surface that is part parabolid above the plane 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 parabolid intersects a plane .
Substitute 1 for z in equation (1),
Consider the parametric equations since the orientation of surface is upward.
Write the expression for vector function .
Substitute for x, for y, 1 for z,
Differentiate the expression with respect to t.
Find the value of .
Find the value of .
Write the expression for .
Substitute for ,
Find the value of by using equation (2).
The expression for surface S, in polar coordinate system is,
Hence the equation is in the form of
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