<< Chapter < Page Chapter >> Page >

[T] Use a CAS and Stokes’ theorem to evaluate S ( curl F · N ) d S , where F ( x , y , z ) = x 2 y i + x y 2 j + z 3 k and C is the curve of the intersection of plane 3 x + 2 y + z = 6 and cylinder x 2 + y 2 = 4 , oriented clockwise when viewed from above.

Got questions? Get instant answers now!

[T] Use a CAS and Stokes’ theorem to evaluate S curl F · d S , where F ( x , y , z ) = ( sin ( y + z ) y x 2 y 3 3 ) i + x cos ( y + z ) j + cos ( 2 y ) k and S consists of the top and the four sides but not the bottom of the cube with vertices ( ±1 , ±1 , ±1 ) , oriented outward.

S curl F · d S = 2.6667

Got questions? Get instant answers now!

[T] Use a CAS and Stokes’ theorem to evaluate S curl F · d S , where F ( x , y , z ) = z 2 i 3 x y j + x 3 y 3 k and S is the top part of z = 5 x 2 y 2 above plane z = 1 , and S is oriented upward.

Got questions? Get instant answers now!

Use Stokes’ theorem to evaluate S ( curl F · N ) d S , where F ( x , y , z ) = z 2 i + y 2 j + x k and S is a triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) with counterclockwise orientation.

S ( curl F · N ) d S = 1 6

Got questions? Get instant answers now!

Use Stokes’ theorem to evaluate line integral C ( z d x + x d y + y d z ) , where C is a triangle with vertices (3, 0, 0), (0, 0, 2), and (0, 6, 0) traversed in the given order.

Got questions? Get instant answers now!

Use Stokes’ theorem to evaluate C ( 1 2 y 2 d x + z d y + x d z ) , where C is the curve of intersection of plane x + z = 1 and ellipsoid x 2 + 2 y 2 + z 2 = 1 , oriented clockwise from the origin.

A diagram of an intersecting plane and ellipsoid in three dimensional space. There is an orange curve drawn to show the intersection.

C ( 1 2 y 2 d x + z d y + x d z ) = π 4

Got questions? Get instant answers now!

Use Stokes’ theorem to evaluate S ( curl F · N ) d S , where F ( x , y , z ) = x i + y 2 j + z e x y k and S is the part of surface z = 1 x 2 2 y 2 with z 0 , oriented counterclockwise.

Got questions? Get instant answers now!

Use Stokes’ theorem for vector field F ( x , y , z ) = z i + 3 x j + 2 z k where S is surface z = 1 x 2 2 y 2 , z 0 , C is boundary circle x 2 + y 2 = 1 , and S is oriented in the positive z -direction.

S ( curl F · N ) d S = −3 π

Got questions? Get instant answers now!

Use Stokes’ theorem for vector field F ( x , y , z ) = 3 2 y 2 i 2 x y j + y z k , where S is that part of the surface of plane x + y + z = 1 contained within triangle C with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1), traversed counterclockwise as viewed from above.

Got questions? Get instant answers now!

A certain closed path C in plane 2 x + 2 y + z = 1 is known to project onto unit circle x 2 + y 2 = 1 in the xy -plane. Let c be a constant and let R ( x , y , z ) = x i + y j + z k . Use Stokes’ theorem to evaluate C ( c k × R ) · d S .

C ( c k × R ) · d S = 2 π c

Got questions? Get instant answers now!

Use Stokes’ theorem and let C be the boundary of surface z = x 2 + y 2 with 0 x 2 and 0 y 1 , oriented with upward facing normal. Define

F ( x , y , z ) = [ sin ( x 3 ) + x z ] i + ( x y z ) j + cos ( z 4 ) k and evaluate C F · d S .
Got questions? Get instant answers now!

Let S be hemisphere x 2 + y 2 + z 2 = 4 with z 0 , oriented upward. Let F ( x , y , z ) = x 2 e y z i + y 2 e x z j + z 2 e x y k be a vector field. Use Stokes’ theorem to evaluate S curl F · d S .

S curl F · d S = 0

Got questions? Get instant answers now!

Let F ( x , y , z ) = x y i + ( e z 2 + y ) j + ( x + y ) k and let S be the graph of function y = x 2 9 + z 2 9 1 with z 0 oriented so that the normal vector S has a positive y component. Use Stokes’ theorem to compute integral S curl F · d S .

Got questions? Get instant answers now!

Use Stokes’ theorem to evaluate F · d S , where F ( x , y , z ) = y i + z j + x k and C is a triangle with vertices (0, 0, 0), (2, 0, 0) and ( 0 , −2 , 2 ) oriented counterclockwise when viewed from above.

F · d S = −4

Got questions? Get instant answers now!

Use the surface integral in Stokes’ theorem to calculate the circulation of field F , F ( x , y , z ) = x 2 y 3 i + j + z k around C , which is the intersection of cylinder x 2 + y 2 = 4 and hemisphere x 2 + y 2 + z 2 = 16 , z 0 , oriented counterclockwise when viewed from above.

A diagram in three dimensions of a vector field and the intersection of a sylinder and hemisphere. The arrows are horizontal and have negative x components for negative y components and have positive x components for positive y components. The curve of intersection between the hemisphere and cylinder is drawn in blue.
Got questions? Get instant answers now!

Use Stokes’ theorem to compute S curl F · d S , where F ( x , y , z ) = i + x y 2 j + x y 2 k and S is a part of plane y + z = 2 inside cylinder x 2 + y 2 = 1 and oriented counterclockwise.

A diagram of a vector field in three dimensional space showing the intersection of a plane and a cylinder. The curve where the plane and cylinder intersect is drawn in blue.

S curl F · d S = 0

Got questions? Get instant answers now!

Use Stokes’ theorem to evaluate S curl F · d S , where F ( x , y , z ) = y 2 i + x j + z 2 k and S is the part of plane x + y + z = 1 in the positive octant and oriented counterclockwise x 0 , y 0 , z 0 .

Got questions? Get instant answers now!

Let F ( x , y , z ) = x y i + 2 z j 2 y k and let C be the intersection of plane x + z = 5 and cylinder x 2 + y 2 = 9 , which is oriented counterclockwise when viewed from the top. Compute the line integral of F over C using Stokes’ theorem.

S curl F · d S = −36 π

Got questions? Get instant answers now!

[T] Use a CAS and let F ( x , y , z ) = x y 2 i + ( y z x ) j + e y x z k . Use Stokes’ theorem to compute the surface integral of curl F over surface S with inward orientation consisting of cube [ 0 , 1 ] × [ 0 , 1 ] × [ 0 , 1 ] with the right side missing.

Got questions? Get instant answers now!

Let S be ellipsoid x 2 4 + y 2 9 + z 2 = 1 oriented counterclockwise and let F be a vector field with component functions that have continuous partial derivatives.

S curl F · N = 0

Got questions? Get instant answers now!

Let S be the part of paraboloid z = 9 x 2 y 2 with z 0 . Verify Stokes’ theorem for vector field F ( x , y , z ) = 3 z i + 4 x j + 2 y k .

Got questions? Get instant answers now!

[T] Use a CAS and Stokes’ theorem to evaluate C F · d S , if F ( x , y , z ) = ( 3 z sin x ) i + ( x 2 + e y ) j + ( y 3 cos z ) k , where C is the curve given by x = cos t , y = sin t , z = 1 ; 0 t 2 π .

C F · d r = 0

Got questions? Get instant answers now!

[T] Use a CAS and Stokes’ theorem to evaluate F ( x , y , z ) = 2 y i + e z j arctan x k with S as a portion of paraboloid z = 4 x 2 y 2 cut off by the xy -plane oriented counterclockwise.

Got questions? Get instant answers now!

[T] Use a CAS to evaluate S curl( F ) · d S , where F ( x , y , z ) = 2 z i + 3 x j + 5 y k and S is the surface parametrically by r ( r , θ ) = r cos θ i + r sin θ j + ( 4 r 2 ) k ( 0 θ 2 π , 0 r 3 ) .

S curl ( F ) · d S = 84.8230

Got questions? Get instant answers now!

Let S be paraboloid z = a ( 1 x 2 y 2 ) , for z 0 , where a > 0 is a real number. Let F = x y , y + z , z x . For what value(s) of a (if any) does S ( × F ) · n d S have its maximum value?

Got questions? Get instant answers now!

For the following application exercises, the goal is to evaluate A = S ( × F ) · n d S , where F = x z , x z , x y and S is the upper half of ellipsoid x 2 + y 2 + 8 z 2 = 1 , where z 0 .

Evaluate a surface integral over a more convenient surface to find the value of A .

A = S ( × F ) · n d S = 0

Got questions? Get instant answers now!

Evaluate A using a line integral.

Got questions? Get instant answers now!

Take paraboloid z = x 2 + y 2 , for 0 z 4 , and slice it with plane y = 0 . Let S be the surface that remains for y 0 , including the planar surface in the xz -plane. Let C be the semicircle and line segment that bounded the cap of S in plane z = 4 with counterclockwise orientation. Let F = 2 z + y , 2 x + z , 2 y + x . Evaluate S ( × F ) · n d S .

A diagram of a vector field in three dimensional space where a paraboloid with vertex at the origin, plane at y=0, and plane at z=4 intersect. The remaining surface is the half of a paraboloid under z=4 and above y=0.

S ( × F ) · n d S = 2 π

Got questions? Get instant answers now!

For the following exercises, let S be the disk enclosed by curve

C : r ( t ) = cos φ cos t , sin t , sin φ cos t , for 0 t 2 π , where 0 φ π 2 is a fixed angle.

What is the length of C in terms of φ ?

Got questions? Get instant answers now!

What is the circulation of C of vector field F = y , z , x as a function of φ ?

C = π ( cos φ sin φ )

Got questions? Get instant answers now!

For what value of φ is the circulation a maximum?

Got questions? Get instant answers now!

Circle C in plane x + y + z = 8 has radius 4 and center (2, 3, 3). Evaluate C F · d r for F = 0 , z , 2 y , where C has a counterclockwise orientation when viewed from above.

C F · d r = 48 π

Got questions? Get instant answers now!

Velocity field v = 0 , 1 x 2 , 0 , for | x | 1 and | z | 1 , represents a horizontal flow in the y -direction. Compute the curl of v in a clockwise rotation.

Got questions? Get instant answers now!

Evaluate integral S ( × F ) · n d S , where F = x z i + y z j + x y e z k and S is the cap of paraboloid z = 5 x 2 y 2 above plane z = 3 , and n points in the positive z -direction on S .

S ( × F ) · n = 0

Got questions? Get instant answers now!

For the following exercises, use Stokes’ theorem to find the circulation of the following vector fields around any smooth, simple closed curve C.

F = y 2 z 3 , z 2 x y z 3 , 3 x y 2 z 2

0

Got questions? Get instant answers now!
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask