- Determine whether the vector field
is conservative. If it is conservative, find a potential function.

- Determine whether the vector field
is conservative. If it is conservative, find a potential function.

- Determine whether the vector field
is conservative. If it is conservative, find a potential function.

- Find scalar potentials for the following vector fields if it is possible to do so. If it is
not possible to do so, explain why.
- If a vector field is not conservative on the set U, is it possible the same vector field could be conservative on some subset of U? Explain and give examples if it is possible. If it is not possible also explain why.
- Prove that if a vector field F has a scalar potential, then it has infinitely many scalar potentials.
- Here is a vector field:F ≡. Find ∫
_{C}F⋅dR where C is a curve which goes fromto. - Here is a vector field:F ≡. Find ∫
_{C}F⋅dR where C is a curve which goes fromto. - Find ∫
_{∂U}F⋅dR where U is the setand F=. - Find ∫
_{∂U}F⋅dR where U is the setand F=. - Find ∫
_{∂U}F⋅dR where U is the setand F=. - Find ∫
_{∂U}F⋅dR where U is the setand F=. - Show that for many open sets in ℝ
^{2}, Area of U = ∫_{∂U}xdy, and Area of U = ∫_{∂U}− ydx and Area of U =∫_{∂U}− ydx + xdy. Hint: Use Green’s theorem. - Two smooth oriented surfaces, S
_{1}and S_{2}intersect in a piecewise smooth oriented closed curve C. Let F be a C^{1}vector field defined on ℝ^{3}. Explain why ∫_{S1}curl⋅ndS = ∫_{S2}curl⋅ndS. Here n is the normal to the surface which corresponds to the given orientation of the curve C. - Show that curl= ∇ψ×∇ϕ and explain why ∫
_{S}∇ψ×∇ϕ⋅ndS = ∫_{∂S}⋅dr. - Find a simple formula for div where α ∈ ℝ.
- Parametric equations for one arch of a cycloid are given by x = aand y = awhere here t ∈. Sketch a rough graph of this arch of a cycloid and then find the area between this arch and the x axis. Hint: This is very easy using Green’s theorem and the vector field F =.
- Let r=where t ∈. Sketch this curve and find the area enclosed by it using Green’s theorem.
- Verify that Green’s theorem can be considered a special case of Stoke’s theorem.
- Consider the vector field = F. Show that ∇× F = 0 but that for the closed curve, whose parametrization is R=for t ∈, ∫
_{C}F⋅dR≠0. Therefore, the vector field is not conservative. Does this contradict Theorem 28.4.8? Explain. - Let x be a point of ℝ
^{3}and let n be a unit vector. Let D_{r}be the circular disk of radius r containing x which is perpendicular to n. Placing the tail of n at x and viewing D_{r}from the point of n, orient ∂D_{r}in the counter clockwise direction. Now suppose F is a vector field defined near x. Show that curl⋅ n = lim_{r→0}∫_{∂Dr}F⋅dR. This last integral is sometimes called the circulation density of F. Explain how this shows that curl⋅ n measures the tendency for the vector field to “curl” around the point, the vector n at the point x. - The cylinder x
^{2}+ y^{2}= 4 is intersected with the plane x + y + z = 2. This yields a closed curve C. Orient this curve in the counter clockwise direction when viewed from a point high on the z axis. Let F =. Find ∫_{C}F⋅dR. - The cylinder x
^{2}+ 4y^{2}= 4 is intersected with the plane x + 3y + 2z = 1. This yields a closed curve C. Orient this curve in the counter clockwise direction when viewed from a point high on the z axis. Let F =. Find ∫_{C}F⋅dR. - The cylinder x
^{2}+ y^{2}= 4 is intersected with the plane x + 3y + 2z = 1. This yields a closed curve C. Orient this curve in the clockwise direction when viewed from a point high on the z axis. Let F =. Find ∫_{C}F⋅dR. - Let F =. Find the surface integral ∫
_{S}curl⋅ndA where S is the surface z = 4 −, z ≥ 0. - Let F =. Find the surface integral ∫
_{S}curl⋅ndA where S is the surface z = 16 −, z ≥ 0. - The cylinder z = y
^{2}intersects the surface z = 8 − x^{2}− 4y^{2}in a curve C which is oriented in the counter clockwise direction when viewed high on the z axis. Find ∫_{C}F⋅dR if F =. - Tell which open sets are simply connected. The inside of a car radiator, A donut.,
The solid part of a cannon ball which contains a void on the interior. The inside of
a donut which has had a large bite taken out of it, All of ℝ
^{3}except the z axis, All of ℝ^{3}except the xy plane. - Let P be a polygon with vertices ,,,,encountered as you move over the boundary of the polygon which is assumed a simple closed curve in the counter clockwise direction. Using Problem 13, find a nice formula for the area of the polygon in terms of the vertices.
- Here is a picture of two regions in the plane, U
_{1}and U_{2}. Suppose Green’s theorem holds for each of these regions. Explain why Green’s theorem must also hold for the region which lies between them if the boundary is oriented as shown in the picture. - Here is a picture of a surface which has two bounding curves oriented as
shown. Explain why Stoke’s theorem will hold for such a surface and sketch
a region in the plane which could serve as a parameter domain for this
surface.

The Mathematical Theory Of Determinants

Download PDFView PDF