Using Stokes’ Theorem to evaluate a surface integral Evaluate∫∫S (∇ x F) # n dS, where F = -y i + x j + z k, in the following cases.a. S is the part of the paraboloid z = 4 - x2 - 3y2 that lies within the paraboloid z = 3x2 + y2 (the blue surface as shown). Assume n pointsin the upward direction on S.b. S is the part of the paraboloid z = 3x2 + y2 that lies within the paraboloidz = 4 - x2 - 3y2, with n pointing in the upward direction on S.c. S is the surface in part (b), but n pointing in the downward direction on S. Outward normalZAvector forExample 3az= 4 - x2 – 3y2nInward normalvector forExample 3bC.Outward normalvector forz = 3x2 + y2Example 3cn

As per the question we are given two paraboloid surfaces S1 : z = 4 - x2 - 3y2 , S2 : z = 3x2 +…

Answered: Outward normal ZA vector for Example 3a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Using Stokes’ Theorem to evaluate a surface integral Evaluate
∫∫_S (∇ x F) # n dS, where F = -y i + x j + z k, in the following cases.
a. S is the part of the paraboloid z = 4 - x² - 3y² that lies within the paraboloid z = 3x² + y² (the blue surface as shown). Assume n points
in the upward direction on S.
b. S is the part of the paraboloid z = 3x² + y² that lies within the paraboloid
z = 4 - x² - 3y², with n pointing in the upward direction on S.
c. S is the surface in part (b), but n pointing in the downward direction on S.

Outward normal
ZA
vector for
Example 3a
z= 4 - x2 – 3y2
n
Inward normal
vector for
Example 3b
C.
Outward normal
vector for
z = 3x2 + y2
Example 3c
n

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