We first briefly review an application of line integrals of vector fields. In particular, we focus on the concept of work [math]W[/math] done by a variable force [math]\vec{F}\left(x,y,z\right)=\langle M\left(x,y,z\right),N\left(x,yz\right),P\left(x,y,z\right)\rangle[/math] along a path [math]C[/math]. [br][br]To compute work done by a force field [math]\vec{F}\left(x,y,z\right)[/math], we: [br][list=1][*]Divide the path [math]C[/math] into [math]n[/math] subarcs. See figure (1).[/*][*]Approximate the work done along the [math]k-[/math]th subarc as [math]\vec{F}\left(x_k,y_k,z_k\right)\cdot\vec{T}\Delta s_k[/math], where [math]\vec{T}_k[/math] is the unit tangent vector to the [math]k-[/math]th subarc and [math]\Delta s_k[/math] is the length of the subarc. [/*][*]Then, we sum up the work done along all the subarcs to find the approximate work done as [math]\sum_{k=1}^n\vec{F}\left(x_k,y_k,z_k\right)\cdot\vec{T}\Delta s_k[/math].[/*][*]Finally, we take the limit of number of subarcs [math]n[/math] approaching [math]\infty[/math]: [/*][/list][center] [math]W=\lim_{n\to\infty}\sum_{k=1}^n\vec{F}\left(x_k,y_k,z_k\right)\cdot\vec{T}\Delta s_k=\int_C\vec{F}\left(x,y,z\right)\cdot\vec{T}ds=\int_{t=a}^{t=b}\vec{F}\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)\cdot\vec{v}dt=\int_C\vec{F}\left(x,y,z\right)\cdot\vec{dr}=\int_CMdx+Ndy+Pdz,[/math] [/center]where [math]ds=\parallel\vec{v}\parallel dt[/math], [math]\vec{v}=\vec{\frac{dr}{dt}}[/math], [math]\vec{T}=\frac{\vec{v}}{\parallel\vec{v}\parallel}[/math], and [math]\vec{F}\left(x,y,z\right)=\langle M\left(x,y,z\right),N\left(x,yz\right),P\left(x,y,z\right)\rangle[/math].
In figure (2), determine whether the work done by the force field along the two paths [math]C_1[/math] and [math]C_2[/math] is positive, negative, or zero.
In figure (3), determine whether the work done by the force field along the two paths [math]C_1[/math] and [math]C_2[/math] is positive, negative, or zero.
[list=1][*]If the result of the line integral [math]\int_C\vec{F}\cdot\vec{T}ds[/math] from point [i]A[/i] to point [i]B [/i]is the same for all possible paths[i], [/i]then the line integral is said to be [b]path-independent [/b]and the field [math]\vec{F}[/math] is said to be[b] [/b][b]conservative.[/b][/*][*][b][/b]If a vector field [math]\vec{F}[/math] can be written as [math]\vec{F}=\vec{\nabla f}[/math], where [math]f[/math] is a scalar-valued function, then [math]f[/math] is a [b]potential function. [/b] [/*][*]There exists a potential function [math]f[/math] such that [math]\vec{F}=\vec{\nabla f}[/math] if and only if the field [math]\vec{F}[/math] is conservative and the line integral [math]\int_C\vec{F}\cdot\vec{T}ds[/math] is path-independent. Also, the line integral can be computed as[b] [/b][math]\int_C\vec{F}\cdot\vec{T}ds=f\left(B\right)-f\left(A\right)[/math].[/*][*]A vector field [math]\vec{F}[/math] is conservative if and only if [math]\oint_C\vec{F}\cdot\vec{T}ds=0[/math] for every closed curve [math]C[/math]. [/*][/list]
Do the vector fields in figures (2) and (3) appear to be conservative or not? Justify your answer in each case.
[list=1][*]Use the parametrized paths to compute the work done by the force fields from point A to point B as shown in figures (4) and (5). [/*][*]In figures (4) and (5), is it possible to use the fundamental theorem of line integrals to compute the work done by the force fields from point A to point B? If so, use the fundamental theorem of line integrals to verify your answers to the previous part. Remember to fully justify your answers. [/*][/list]
Compute the line integral of the vector field along the line segment shown in figure (6).
Compute the line integral of the vector field from point A to point B as shown in figure (7), where the curve is parametrized as [math]\vec{r}\left(t\right)=\left\langle\frac{\ln\left(t\right)}{\ln\left(2\right)},t^{\frac{3}{2}},t\cos\left(\pi t\right)\right\rangle[/math].
Did you notice a relation between the answers to questions (5) and (6)? Did you expect that to be the case?
Evaluate the line integral [math]\int_C2xye^{-yz}dx+\left(x^2e^{-yz}-x^2yze^{-yz}\right)dy-x^2y^2e^{-yz}dz[/math], where [math]C[/math] is a curve from (0,0,1) to (1,0,1).