Question

(a) Let
F = (4x + 3y) i  + (3y + 8z) j  + (8z + 4x)k


Let C be the circumference of the circle of radius 8 in the plane x + y + z = 5, oriented counterclockwise as viewed from the point (5, 5, 5).
Find integral_C F · dr =


(b) Now suppose we know
G = (4x^2 + 3y^2) i  + (3y^2 + 8z^2) j  + (8z^2 + 4x^2)k
but only very near to the point (5, 5, 5).

Let R be a regular octagon with area 0.08 in the plane y = 5 and C be its perimeter oriented counterclockwise when viewed from (0, 10, 0).


Estimate
integral_C G · dr≈ Give your answer to 3 decimal places.

Answer 1

@ganeshie8 do you think the best way to find the integral is to use parametric eqn?

Answer 2

or can I use stokes' thm?

Answer 3

you have a closed path so yes you can use stokes

Answer 4

ok I get a little confuse with stokes thm, do i just find the curl and multiply it by the area of the circle? @ganeshie8

Answer 5

what does stokes thm say ?

Answer 6

@ganeshie8 is helping you he's really good at math!

Answer 7

curlF dot dA

Answer 8

I know he is :)

Answer 9

curlF= -8, -4, -3

Answer 10

\[\oint_C \vec{F}\cdot d\vec{r} = \iint _S \mathrm{curl}(\vec{F} )\cdot \hat{n} dS\]

Answer 11

find ndS and take the dot product with curl

Answer 12

whats ndS? the normal?

Answer 13

yes normal to the plane

Answer 14

so <5,5,5>

Answer 15

\[\hat{n}dS = \dfrac{\vec{N}}{\vec{N}\cdot \hat{k}}dxdy\]

Answer 16

\[\hat{n}dS = \dfrac{\langle 5,5,5 \rangle }{\langle 5,5,5 \rangle\cdot \langle 0,0,1 \rangle}dxdy\]

Answer 17

so I don't need the area of the circle pi(8)^2?

Answer 18

\[\hat{n}dS = \langle 1,1,1 \rangle dxdy\]

Answer 19

so its just 5?

Answer 20

plug that in stokes thm formula

Answer 21

i'm confuse, how do you ge <1,1,1>

Answer 22

I'm getting -15 @ganeshie8

Answer 23

do i multiply -15 times pi(8)^2

Answer 24

\[\begin{align}\oint_C \vec{F}\cdot d\vec{r} &= \iint _S \mathrm{curl}(\vec{F} )\cdot \hat{n} dS \\~\\
&= \iint_S \langle -8,-4,-3 \rangle \cdot \langle 1,1,1 \rangle dxdy \\~\\
&= \iint_S -15 dxdy \\~\\
&= -15\iint_S 1 dxdy \\~\\

\end{align}\]

Answer 25

so 1dxdy is the area? @ganeshie8

Answer 26

yes but it is the area of projection of circular disk onto the xy plane

Answer 27

\[\begin{align}\oint_C \vec{F}\cdot d\vec{r} &= \iint _S \mathrm{curl}(\vec{F} )\cdot \hat{n} dS \\~\\
&= \iint_S \langle -8,-4,-3 \rangle \cdot \langle 1,1,1 \rangle dxdy \\~\\
&= \iint_S -15 dxdy \\~\\
&= -15\iint_S 1 dxdy \\~\\
&= -15 (\text{Area of projection of circular disk onto xy plane })\\~\\
&= -15(\pi \times 8^2 \times \frac{1}{\sqrt{3}})
\end{align}\]

Answer 28

you dont have answers for these problems ?

Answer 29

i would like @eliassaab sir to double my work

Answer 30

or @Kainui @SithsAndGiggles

Answer 31

no I don't :/ and I can't try it out because it was due tusday

Answer 32

its webassign right ?

Answer 33

yeah

Answer 34

this problem its hard, I have like no idea what to do for part b @ganeshie8

Answer 35

part b should be easy if you understand part a

Answer 36

for part b :
find the curl of G at (5,5,5) and dot with ndS

Answer 37

so I find the curl first and than sub in (5,5,5)?

Answer 38

cool tha's what I got, i just need to multiply it by (0.08) ? @ganeshie8

Answer 39

Yes !

Answer 40

G = (4x^2 +3y^2)i+(3y^2+ 8z^2)j+(8z^2+ 4x^2)k

curl(G) = <-16z, -8x, -6y>
curl(G) at (5,5,5) = <-80, -40, -30>

\[\int_C \vec{G}\cdot d\vec{r} = \iint_S \mathrm{curl}(G) \cdot \hat{n} dS \approx \iint_S \langle -80,-40,-30\rangle \cdot \langle 0,1,0\rangle dxdz\]

Answer 41

so just just (-40) * (.08) ? @ganeshie8

Answer 42

looks good!

Answer 43

cool

Answer 44

-3.200

Answer 45

wondering why they would ask for 3 decimal places

Answer 46

yeah let me know once you have the answers

Answer 47

i get -3.2

Answer 48

@ganeshie8 to bad that I can't check the answer :/

Answer 49

should the answer be negative?

Answer 50

yes our orientation and normal vector look good

Answer 51

I guess its time to more to a new question haha thank you so much for your help :) you are the best