Chapter 1. calc_tutorial_17_2_020

1.1 Problem Statement

{7}
{4}
2*$a
$a*$b

Let F = <−z2, 2zx, $ay − x2> and let C be a simple closed curve in the plane x + y + z = r, r = $a that encloses a region of area $b√3 (see figure below). Calculate , where C is oriented in the counterclockwise direction (when viewed from above the plane).

1.2 Step 1

Question Sequence

Question 1.1

Recall Stokes' Theorem. Assume that S is an oriented surface with parametrization G: D → S, where D is a domain in the plane bounded by smooth, simple closed curves, and G is one-to-one and regular, except possibly at the boundary of D. More generally, S may be a finite union of surfaces of this type. Then Stokes' Theorem gives us the following equation.

The integral on the left is defined relative to the boundary orientation of ∂S. If S is closed (that is, ∂S is empty), then the surface integral on the right is v4/Yoxb7daTg0BC3/fg8y28QBiU=.

This equation also introduces the notation curl(F). Given a vector field F = <F1, F2, F3> and letting the table below represent the matrix A, we define curl(F) as follows.

i j k
(∂ / ∂x) (∂ / ∂y) (∂ / ∂z)
F 1 F 2 F 3

curl(F) = det(A) = <(∂F3 / ∂y) − (∂F2 / ∂z), (∂F1 / ∂z) − (∂F3 / ∂x), (∂F2 / ∂x) − (∂F1 / ∂y)>

Compute the curl of F = <−z2, 2zx, $ay − x2>.

curl(F) =

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

We will denote S to be the region enclosed by C. Thus ∂S = C and we can rewrite Stokes' Theorem for this problem as follows.

Without more specific details about the curve C, we cannot calculate directly. However, we will be able to use Stokes' Theorem to calculate instead.

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

To compute the flux of the curl through the surface, we will use the following Vector Surface Integral equation.

Parametrize the plane x + y + z = $a using the parameters x and y.

G(x, y) = <x, y, nc1ItEz0kR4= - x - y>

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Use the parametrization of the plane, G(x, y) = <x, y, $a − x − y>, to compute n(x, y) = Tx × Ty, the normal vector pointing outward through the top of the plane. Given the counterclockwise orientation of C when viewed from above, this is the appropriate normal vector we need to calculate the integral.

Tx = (∂G / ∂x)

= <0VV1JcqyBrI=, 1Wh3cvJ2xF4=, UYinCekNO0E=>

Ty = (∂G / ∂y)

= <1Wh3cvJ2xF4=, 0VV1JcqyBrI=, UYinCekNO0E=>

n(x, y) = Tx × Ty

= <0VV1JcqyBrI=, 0VV1JcqyBrI=, 0VV1JcqyBrI=>

Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

Use the parametrization of the plane and the expression for curl(F) found in Step 2 to calculate curl(F(G(x, y))).

curl(F) = <$a − 2x, 2x − 2z, 2z>

curl(F(G(x, y))) = <$a - 2(x), 2(x) - XvVM00l89Is=($a - x - y), 2(nc1ItEz0kR4= - x - y)>

= <$a - 2x, h4XZagboIgc= + 2y - $c, SFgqQUkJGdg= - 2x - 2y>

Correct.
Incorrect.

1.6 Step 5

Question Sequence

Question 1.5

Compute the dot product curl(F(G(x, y))) · n(x, y).

curl(F(G(x, y))) · n(x, y) = <$a − 2x, 4x + 2y − $c, $c − 2x − 2y> · <1, 1, 1>

= nc1ItEz0kR4=

Correct.
Incorrect.

1.7 Step 6

Question Sequence

Question 1.6

Compute .

= $a · Area(D)

Recall that D comes from our parametrization of the surface S. S is an oriented surface with parametrization G: D → S, where D is a domain in the plane bounded by smooth, closed curves. Therefore Area(D) is the area of the base in the xy-plane which lies underneath the surface S bounded by C.

To calculate Area(D), we must use Area(S) = $b√3 and work backwards. We use the following general equation for surface area.

Area(S) =

Fortunately, the region S lies in a plane, so the normal vector is n(x, y) = <1, 1, 1> at every point on the plane G(x, y) = <x, y, $a − x − y>, and thus its norm will be a constant.

Therefore, ||n(x, y)|| =

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

Substituting these values into the surface area equation to solve for Area(D).

Therefore, Area(D) = iSba6t70dtA=.

Correct.
Incorrect.

1.8 Step 7

Question Sequence

Question 1.7

Calculate .

= U2GIbglD1oM=

Correct.
Incorrect.