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Vector Calculus: Three Dimensional Space

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Note that in 4, I give a review and classify two of the given sets. I think you can do the rest if you read the review. Should you have further questions you can contact me.

2. Let us make a connection with the first exercise since they are giving us the hint. What can we say about the curves? Well, we can look at the starting and ending points $x_0$ and $x_1$, respectively:
item[(1)] For exercise 1 they are $mathbf{x}_0=(-1,1,-1)$ and $mathbf{x}_1=(1,1,1)$.
item[(A)] For $Gamma_A$, $mathbf{x}_0=(-1,1,-1)$ and $mathbf{x}_1=(1,1,1)$.
item[(B)] For $Gamma_B$, $mathbf{x}_0=(0,0,0)$ and $mathbf{x}_1=(2,4,8)$.
item[(C)] For $Gamma_C$, $mathbf{x}_0=(1,-1,-1)$ and $mathbf{x}_1=(1,-1,-1)$.
item[(D)] For $Gamma_D$, $mathbf{x}_0=(1,1,1)$ and $mathbf{x}_1=(-1,1,-1)$.

How can this help us? we now look at the vector field $mathbf{F}=y^2mathbf{i}+2xymathbf{j}$. Note that it is conservative by the ``cross derivative test'' (Clairauts' Theorem). Therefore a line integral from a point $textbf{x}_0$ to $mathbf{x}_1$ is independent of the path.

We know that the answer to exercise 1 is $2$, therefore the line integral of $mathbf{F}$ along $Gamma_A$ equals $2$. Note that
$Gamma_D$ is traced backwards when compared to exercise 1, but the endpoints are the same, therefore the line integral of
$mathbf{F}$ along $Gamma_D$ equals $-2$. Now for $Gamma_C$ note that the starting and ending points are the same and since
$mathbf{F}$ is conservative, the line integral of $mathbf{F}$ along $Gamma_C$ equals $0$. We are left with $Gamma_B$ and
there is one more value left, so the line integral of $mathbf{F}$ along $Gamma_B$ equals $32$ (please check it by hand!).

4. Throughout this section we consider a continuous (or better, differentiable) function of three variables $F:R^3toR$ (also
called scalar field) and a value ...

Solution Summary

This is the answer to the line integral of a vector field in three dimensional space we are able to solve for the value of different line integrals of that same vector field without the need for explicit calculations. For this we use that we deal with a conservative vector field.

We classify given subsets in three dimensional space as being: a path of a curve, a path of a loop, a two dimensional (flat) region, a graph surface, the boundary of a domain, the level set of a scalar field, an oriented surface, a domain.