- Let r=for t ∈. Find the length of this curve.
- Let r=for t ∈. Find the length of this curve.
- Let r=for t ∈. Find the length of this curve.
- Suppose for t ∈ the position of an object is given by r= ti + cosj + sink. Also suppose there is a force field defined on ℝ
^{3},F≡ 2xyi +j + y^{2}k. Find the work ∫_{C}F⋅dR where C is the curve traced out by this object having the orientation determined by the direction of increasing t. - In the following, a force field is specified followed by the parametrization of a
curve. Find the work.
- F =,r=,t ∈
- F =,r=,t ∈
- F =,r=,t ∈
- F =,r=,t ∈

- F =
- The curve consists of straight line segments which go from toand finally to. Find the work done if the force field is
- F =
- F =
- F =
- F =

- F =
^{∗}Recall the notion of the gradient in Definition 20.3.5 on Page 1314. Show the vector fields in the preceding problems are respectivelyand ∇

. Thus each of these vector fields is of the form ∇f where f is a function of three variables. For each f in the above, compute f−fand compare with your solutions to the above line integrals. You should get the same thing from f− f. This is not a coincidence and will be fully discussed later. Such vector fields are called conservative.- Here is a vector field and here is the parametrization of a curve C. R=where t goes from 0 to π∕4. Find ∫
_{C}F⋅dR. - If f and g are both increasing functions, show that f ∘ g is an increasing function also. Assume anything you like about the domains of the functions.
- Suppose for t ∈the position of an object is given by r= ti + tj + tk. Also suppose there is a force field defined on ℝ
^{3},F≡ yzi + xzj + xyk. Find ∫_{C}F ⋅dR where C is the curve traced out by this object which has the orientation determined by the direction of increasing t. Repeat the problem for r= ti + t^{2}j + tk. - Suppose for t ∈the position of an object is given by r= ti + tj + tk. Also suppose there is a force field defined on ℝ
^{3},F≡ zi + xzj + xyk. Find ∫_{C}F ⋅ dR where C is the curve traced out by this object which has the orientation determined by the direction of increasing t. Repeat the problem for r= ti + t^{2}j + tk. - Let Fbe a given force field and suppose it acts on an object having mass m on a curve with parametrization,for t ∈. Show directly that the work done equals the difference in the kinetic energy. Hint:
etc.

- Suppose for t ∈the position of an object is given by
Also suppose there is a force field defined on ℝ

^{3},Find the work ∫

_{C}F ⋅dR where C is the curve traced out by this object which has the orientation determined by the direction of increasing t. - Here is a vector field and here is the parametrization of a curve C. R=where t goes from 0 to π∕4. Find ∫
_{C}F⋅dR. - Suppose for t ∈the position of an object is given by r= ti + tj + tk. Also suppose there is a force field defined on ℝ
^{3},F≡ yzi + xzj + xyk. Find ∫_{C}F ⋅dR where C is the curve traced out by this object which has the orientation determined by the direction of increasing t. Repeat the problem for r= ti + t^{2}j + tk.You should get the same answer in this case. This is because the vector field happens to be conservative. (More on this later.)

- Suppose F= ∇ϕin some open set containing a piecewise smooth curve C oriented to go from p to q. Then use the chain rule to verify that ∫
_{C}F⋅dR = ϕ− ϕ. This illustrates part of the fundamental theorem of line integrals which will be presented later. Note that this says the line integral is path independent because computing it did not depend on the particular parametrization for C, only on the end points and orientation of the curve.

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