Problem 1
Each of the figures below contains a sketch of a surface and its boundary . Stokes' theorem says that if is a correctly oriented unit normal vector to . Add to each sketch a typical such normal vector.
Problem 2
Let be a finite region in the -plane whose boundary consists of a single piecewise smooth, simple closed curve oriented consistently with (so that if you walk along in the direction of the arrow, is on your left). Let and have continuous first partial derivatives at every point of . Use Stokes' theorem to show that
i.e. to show Green's theorem.
Problem 3
Verify the identity for any continuously differentiable scalar fields and and curve that is the boundary of a piecewise smooth surface.
Problem 4
Let be the curve of intersection of the cylinder and the surface , oriented anticlockwise as seen from . Let . Calculate :
(a). By Direct Evaluation, and
(b). By Using Stokes' Theorem
Problem 5
Evaluate where and is the curve
Problem 6
Find the value of where and is the hemisphere
oriented so that the surface normals point away from the centre of the hemisphere.
Problem 7
Let be the part of the surface which lies above the -plane. Let . Calculate where is the upward normal on .
Problem 8
Let be the intersection of the paraboloid with the cylinder , oriented anticlockwise when viewed from high on the -axis. Let . Find .
Problem 9
Let , and let be the part of the surface that lies above the square , in the -plane. Find the flux of upward through .
Problem 10
Evaluate the integral , in which and is the triangular path from to to to .
Problem 11
Let . Use Stokes' theorem to evaluate the line integral where is the intersection of the plane and the ellipsoid , oriented anticlockwise when viewed from high on the -axis.
Problem 12
Consider the vector field in .
(a). Compute the line Integral where Consists of Three line Segments: from to , then from to , Finally from to
(b). A Simple Closed Curve Lies on the Plane , Enclosing a Region on the Plane of Area , and Oriented Anticlockwise as Observed from the Positive -axis. Compute the line Integral
Problem 13
Let be the curve given by the union of three parametrised curves:
(a). Draw a Picture of . Clearly Mark Each of the Curves , , and and Indicate the Orientations given by the Parametrisations
(b). Find and Parametrise an Oriented Surface whose Boundary is (with the given orientations)
(c). Compute the line Integral where
Problem 14
We consider the cone with equation . The cone is oriented so that the normal vectors point downwards (and away from the -axis). Let .
(a). Let Be the part of the Cone that Lies between the Planes and . Use Stokes' Theorem to Determine . Make a Sketch Indicating the Orientations of and of the contour(s) of Integration
(b). Let Be the part of the Cone that Lies below the Plane and above . Determine the Flux of across . Justify Your Answer, including a Sketch
Problem 15
Consider the curve that is the intersection of the plane and the cylinder , and suppose is oriented so that it is traversed clockwise as seen from above. Let . Use Stokes' theorem to evaluate .
Problem 16
(a). Consider the Vector Field in . Compute the line Integral , where Consists of Three line Segments: from to , then from to , Finally from to
(b). A Simple Closed Curve Lies in the Plane . the Surface This Curve Surrounds inside the Plane Has Area . the Curve is Oriented Anticlockwise as Observed from the Positive -axis. Compute , where is as in (a)
Problem 17
Evaluate the line integral
where is the curve parametrised by , .
Problem 18
A simple closed curve lies in the plane . The surface this curve surrounds inside the plane has area . The curve is oriented in a clockwise direction as observed from the positive -axis looking down at the plane.
Compute the line integral of around .
Problem 19
Let be the oriented curve consisting of 5 line segments which form the paths from to , from to , from to , from to , and from to . Let
Evaluate the integral .
Problem 20
Suppose the curve is the intersection of the cylinder with the surface , traversed clockwise if viewed from the positive -axis. Evaluate the line integral
Problem 21
Evaluate where is that part of the sphere above the plane , is the upward unit normal, and
Problem 22
Let . Use Stokes' theorem to evaluate along the path consisting of the straight line segments successively joining the points to to to to to , and back to .
Problem 23
Let . Let be the oriented curve consisting of four line segments from to , from to , from to , and from to .
(a). Draw a Picture of . Clearly Indicate the Orientation on Each line Segment
(b). Compute the Work Integral
Problem 24
Evaluate where and is the surface , , and is a unit normal to obeying .
Problem 25
Let be the curved surface , , oriented by the outward normal. Define , where
Find .
Problem 26
Let be a circle of radius lying in the plane . Use Stokes' theorem to calculate the value of where . (You may use either orientation of the circle.)
Problem 27
Let denote the oriented surface consisting of the top and four sides of the cube whose vertices are , oriented outward. If , find the flux of through .
Problem 28
Let denote the part of the spiral ramp (helicoidal surface) parametrised by
Let denote the boundary of with orientation specified by the upward pointing normal on . Find
Problem 29
Let be the intersection of and . The curve is oriented anticlockwise when viewed from high on the -axis. Let
Evaluate .
Problem 30
(a). Find the Curl of the Vector Field
(b). Let Be the Curve in from to , Consisting of Three Consecutive line Segments Connecting to , to , and to . Evaluate where is the Vector Field from (a)
Problem 31
(a). Let Be the Bucket-shaped Surface Consisting of the Cylindrical Surface between and , and the Disc inside the -plane of Radius Centred at the Origin. (The Bucket Has a Bottom but no lid.) Orient so that the Unit Normal Points Outward. Compute the Flux of the Vector Field through , where
(b). Compute the Flux of the Vector Field through , where is as in (a)
Problem 32
Let .
(a). Write down the Domain of
(b). Circle the Correct statement(s): (i) is Connected. (ii) is Simply Connected. (iii) is Disconnected
(c). Compute
(d). Let Be the Square with Corners in the Plane , Oriented Clockwise (viewed from above). Compute
(e). Is Conservative?
Problem 33
A physicist studies a vector field . From experiments, it is known that is of the form
for some real numbers and . It is further known that for some differentiable vector field .
(a). Determine and
(b). Evaluate the Surface Integral where is the part of the Ellipsoid for Which , Oriented so that Its Normal Vector Has a Positive -component
Consider the circle defined by that sits in the -plane
Problem 34
Let be the curve in the -plane from the point to the point consisting of the ten line segments consecutively connecting the points , , , , , , , , , , . Evaluate the line integral where .
Problem 35
Let . Evaluate around the curve of intersection of the cylinder with the surface , traversed anticlockwise as viewed from high on the -axis.
Problem 36
Explain how one deduces the differential form of Faraday's law from its integral form
Problem 37
Let be the curve given by the parametric equations , , , , and let . Use Stokes' theorem to evaluate .
Problem 38
Use Stokes' theorem to evaluate where is the closed curve which is the intersection of the plane with the sphere . Assume that is oriented clockwise as viewed from the origin.
Problem 39
Let be the part of the half cone , , that lies below the plane .
(a). Find a Parametrisation for
(b). Calculate the Flux of the Velocity Field downward through
Problem 40
Consider where is the portion of the sphere that obeys , is the upward pointing normal to the sphere, and .
Find another surface with the property that and evaluate .
The surface is any surface bounded by the intersection of and .
A specific example of is the region on the plane bounded by the circle