The circulation of the vector field F around the triangle is -324.
Stokes' theorem relates the circulation of a vector field around a closed curve to the curl of the vector field over the surface enclosed by the curve.
Therefore, to use Stokes' theorem to find the circulation of the vector field F = 6yi + 7zj + 6xk around the triangle obtained by tracing out the path from (4,0,0) to (4,0,6), to (4,3,6), and back to (4,0,0), we need to find the curl of F and the surface enclosed by the triangle.
The curl of F is given by:
curl F = ∇ x F
= (d/dx)i x (6yi + 7zj + 6xk) + (d/dy)j x (6yi + 7zj + 6xk) + (d/dz)k x (6yi + 7zj + 6xk)
= -6i + 6j + 7k
To find the surface enclosed by the triangle, we can take any surface whose boundary is the triangle.
One possible choice is the surface of the rectangular box whose bottom face is the triangle and whose top face is the plane z = 6.
The normal vector of the bottom face of the box is -xi, since the triangle is in the yz-plane, and the normal vector of the top face of the box is +zk. Therefore, the surface enclosed by the triangle is the union of the bottom face and the top face of the box, plus the four vertical faces of the box.
Applying Stokes' theorem, we have:
∮C F · dr = ∬S curl F · dS
where C is the boundary of the surface S, which is the triangle in this case.
Since the triangle lies in the plane x = 4, we can parameterize it as r(t) = (4, 3t, 6t) for 0 ≤ t ≤ 1.
Then, dr/dt = (0, 3, 6) and we have:
∮C F · dr = [tex]\int 0^1[/tex] F(r(t)) · dr/dt dt
= [tex]\int 0^1[/tex](0, 18y, 42x) · (0, 3, 6) dt
= [tex]\int 0^1[/tex]378x dt
= 378/2
= 189.
On the other hand, the surface S has area 6 x 3 = 18, and its normal vector is +xi, since it points outward from the box.
Therefore, we have:
∬S curl F · dS = ∬S (-6i + 6j + 7k) · xi dA
[tex]= \int 0^6 ∫0^3 (-6i + 6j + 7k) .xi $ dy dx[/tex]
[tex]= \int 0^6 \int 0^3 (-6x) dy dx[/tex]
= -54 x 6
= -324
Thus, we have:
∮C F · dr = ∬S curl F · dS = -324.
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Stokes' theorem relates the circulation of a vector field around a closed path to the curl of the vector field over the surface bounded by that path. The circulation of the given vector field F around the given triangular path can be calculated as follows:
First, we find the curl of the vector field F:
curl(F) = ( ∂Fz/∂y - ∂Fy/∂z )i + ( ∂Fx/∂z - ∂Fz/∂x )j + ( ∂Fy/∂x - ∂Fx/∂y )k
= 6i + 7j + 6k
Next, we find the surface integral of the curl of F over the triangular surface bounded by the given path. The surface normal vector for this surface can be calculated as the cross product of the tangent vectors at two arbitrary points on the surface, say (4,0,0) and (4,0,6):
n = ( ∂r/∂u x ∂r/∂v ) / | ∂r/∂u x ∂r/∂v |
= (-6i + 0j + 4k) / 6
where r(u,v) = <4,0,u+v> is a parameterization of the surface.
Then, the surface integral of the curl of F over the triangular surface can be calculated as:
∫∫(S) curl(F) ⋅ dS = ∫∫(D) curl(F) ⋅ n dA
where D is the projection of the surface onto the xy-plane, which is a rectangle with vertices (4,0), (4,3), (4,6), and (4,0), and dA is the differential area element on D. The circulation of F around the given path is then given by:
∫(C) F ⋅ dr = ∫∫(D) curl(F) ⋅ n dA
= (6i + 7j + 6k) ⋅ (-i/6) (area of D)
= -19/2
Therefore, the circulation of the vector field F around the given triangular path is -19/2.
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A = 5x^2+3xy-7y^2 B = -x^2+2xy+8y^2 B, minus, A, equals
I WILL GIVE BRAINLYIST!!!!!!!!!!! Help!!!!!
Answer: -6x^2-xy+15y^2
Step-by-step explanation:
The value of B-A is -6x²-xy+15y² .
What is an Equation ?A statement where the algebraic expressions are equated with an equal sign with each other is called an Equation.
It is given that
A = 5x²+3xy-7y²
B = -x²+2xy+8y²
To calculate is B - A
B- A = ² (-x²+2xy+8y²) - (5x²+3xy-7y² )
B-A = -x²+2xy+8y² - 5x²- 3xy +7y²
B-A = -6x²-xy+15y²
Therefore the value of B-A is -6x²-xy+15y² .
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Kim got to the store at 11:19 in the morning. She spent 24 minuets in the store how many minuets after she got to the store and left?
Answer:
11:43 am
Step-by-step explanation:
Given data
Time of arrival= 11:19 am
We are told that she spent 24 minutes in the store
Based on this, we can estimate the time she left the store
The time she left
=11:19+ 24
=11:43 am
Hence she left 11:43 am
what is the slope of this line? -2x + 3y =-6
Answer:
Slope: 2/3
Step-by-step explanation:
You have to first write the equation into slope-intercept form...
y = mx + b
So...
-2x + 3y = -6
becomes...
3y = 2x - 6
y = 2/3x - 2
Which means...
2/3 is the slope
Hope this help :)
Let me know if there are any mistakes!!
Can someone help me in this math problem? It’s a picture of the problem if you can!
When a point is reflected across a line of reflection, it remains the same distance from the line.True or False
Answer:
True
Step-by-step explanation:
To answer this question, I will use the following illustrations.
Assume a point (x,y) is reflected across the x-axis.
Using reflection rule, the new point will be: (x,-y)
On a coordinate plane, the x-axis is represented as: (x,0)
So, we will calculate the distance between (x,y) and (x,0) and also calculate the distance between (x,-y) and (x,0) using the following distance formula.
[tex]D= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]
Distance between (x,y) and (x,0)
[tex]D= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]
[tex]D= \sqrt{(x - x)^2 + (0 - y)^2}[/tex]
[tex]D= \sqrt{(0)^2 + (- y)^2}[/tex]
[tex]D= \sqrt{0 + y^2}[/tex]
[tex]D= \sqrt{y^2}[/tex]
[tex]D = y[/tex]
Distance between (x,-y) and (x,0)
[tex]D= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]
[tex]D= \sqrt{(x - x)^2 + (0 - (-y))^2}[/tex]
[tex]D= \sqrt{(0)^2 + (0+y)^2}[/tex]
[tex]D= \sqrt{0 + y^2}[/tex]
[tex]D= \sqrt{y^2}[/tex]
[tex]D = y[/tex]
See that the calculated distance are equal.
Hence, the given statement is true
can someone please help me??
Answer:
There is only one solution and it is the point of intersection
Step-by-step explanation:
Which equation represents how much it would cost for any number of rides? * 1 point r = 1.50c c = r - 1.50 r = 1.50 + c c = 1.50r
50 points for this helpppp plzz
Answer:
ok
Step-by-step explanation:
Answer:
srry I needed the points hope u don\t mind
Step-by-step explanation:
I NEED HELP PLEASE. I WILL GIVE BRAINLIST IF RIGHT!
Answer:
Shift f up by 1 unit
Step-by-step explanation:
The email address lol
if p is the smallest of four integers, what is their sum interms of P
Answer:
If p is the smallest of n consecutive integers of the same sign than we have p , p+1 , p+2 , … , p+(n−1) ,
So the sum is
∑k=0n−1(p+k)=∑k=0n−1p+∑k=0n−1k=np+n2−n2
Here n=4
So we have 4p+6
And checking
p+(p+1)+(p+2)+(p+3)=4p+6
Note if p=−v
Than you have the same thing as if p=v−n+1 just negative for example 3 consecutive integers the smallest is −5 so the sum is −5+(−4)+(−3)=3×−5+32−32=−15+3=−12
On the other hand:
−(3+4+5)=−(3×3+32−32)=−(9+3)=−12
If p=−v the sum of next v+1 integers is −(∑k=0vk)=−(v2+v2)
Than needs an other v integers to bring it up to 0 again. From there it is
∑k=0hk=h2+h2
Where h=n−(2v+1) .
So recap if p is the smallest of n consecutive integers their sum is
p+(p+1)+(p+2)+…+(p+(n−1))=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪np+n2+n2−(n|p|+n2+n2)((n−|p|)2+(n−|p|)2)−(|p|p+|p|2+|p|2)((n−|p|)2+(n−|p|)2)n≥0p<0∧n<|p|+1p<0∧|p|<n<2|p|+1p<0∧n>2|p|+1
Step-by-step explanation:
what is the equivalent Ratios to 18:15
Answer:
6:5 in simplest form
Step-by-step explanation:
Answer:
6:5
Step-by-step explanation:
The greatest common factor, and only shared factor is 3. Use three and divide on both sides to get 6 and 5
PLEASE HELP ME WITH THIS I HATE SCHOOL
Answer:
B.)[tex]\sqrt{245}[/tex]
Step-by-step explanation:
[tex]\sqrt{245}[/tex]=15.65
7[tex]\sqrt{5}[/tex]=15.65
Hope this helps! :)
Answer:
its is B
Step-by-step explanation:
I just put 7/5 and put the answers below in my calculator and got whatever went with the answer when I put 7/5 (I dont have the square root sign)
Since Beth was born, the population of her town has increased at a rate of 850 people per year. On Beth's 9th birthday, the total population was nearly 307,650, Write and solve a linear equation to find the population on Beth's 16th birthday.
I only need the equation pls help
Edit: the equation needs to be in slope intercept form
Answer:
editor edit
37,650
Step-by-step explanation:
37650
Use algebra tiles to find the difference. Let one yellow tile equal +1 and one red tile equal –1. 12 – (–9)
Answer:
21
Step-by-step explanation:
12 - (-9)
Open The bracket :
Recall: - * - = +
Hence,
12 + 9 = 21
7.1 as a percentage
Answer:
its 710%
Step-by-step explanation:
710% hope it helps!
Answer the photo below thanks
Answer:
I think its D
Step-by-step explanation:
find the side of the square whose perimeter 20 m
help me please i have no idea what i’m doing
Answer:
1. 5
2. 5/3
3. 4.5
4. 21
5. 10
6. (-12
7. (-1.5
The standard form of y = 5(x + 4)(x + 5) is
Answer:
mark me as brainliest plz
Answer:
21 is Correct answer hope you will like it
Calculate the unit price of the following item:
Chocolate chip cookies cost $2.79 per package and
there are 30 cookies inside. What is the price per
cookie? *
A typical person takes about 24000 breaths each day. Write an equation you can use to find the number of breaths a typical person takes each minute.
Step-by-step explanation:
Welp if a typical person takes 24000 breaths each day that's means
24000÷24=1000
so that's means 1000 per hour
if a hour take 60 Min that's mean
1000÷60=16.6666667 don't tell me I'm not sure about this answer and not really good at this :p
An arithmetic sequence has second term 28 and the fifth term 52. (3 marks) a. Find u1 and d. b. Find u10
Answer: See explanation
Step-by-step explanation:
Second term = a + d = 28 ....... i
Fifth term = a + 4d = 52 ......... ii
Subtract equation I from ii
3d = 24
d = 24/3 = 8
d = common difference = 8
Since a + d = 28
a + 8 = 28
a = 28 - 8 = 20
First term = a = 20
u10 = a + (n-1)d
= a + (10-1)8
= 20 + (9 × 8)
= 20 + 72
= 92
what's the answer for 1/2 of 3/5?
Answer:
3/10
Step-by-step explanation:
A cake recipe calls for 2 1/4 cups of flour. Mrs Allen measured 3/4 cup of whole wheat flour and plans to use bleached flour for the remainder. How much bleached flour should she use?
Answer:
she should use 1 2/4 cups of bleached flour
Nathan Reynolds bought a car for $15,000. He made a down payment of $3,000. Nathan secured a loan for the balance of the purchase price at 8 percent interest for four years.
determine the monthly payment and total interest
Answer:
1) The monthly payment is approximately $292.96
2) The total interest is approximately $2,061.84
Step-by-step explanation:
1) The financing method Nathan Reynolds applied to buy the car is given as follows;
The purchase price of the car = $15,000
The down payment made for the car = $3,000
The
The interest on the loan he took for the balance of the purchase price, R = 8%
The number of years for which he took the loan, n = 4 years = 48 months
We have;
The balance of the purchase price = (The purchase price of the car) - (The down payment made for the car)
∴ The balance of the purchase price = $15,000 - $3,000 = $12,000
The balance of the purchase price = $12,000 = The loan Nathan secured
The monthly payment on a loan is given by the following formula
[tex]A = P \times \dfrac{r \cdot (1 + r)^n}{(1 + r)^n - 1}[/tex]
Where;
A = The payment amount made monthly
P = The principal or loan amount = $12,000
r = The interest rate divided by 12 = R/12 = 8/12 = 2/3
n = The number of months the monthly payment will be made = 48 months
By plugging in the variable values, we get;
[tex]A = 12,000 \times \dfrac{\dfrac{2}{300} \times \left (1 + \dfrac{2}{300} \right ) ^{48}}{ \left (1 + \dfrac{2}{300} \right )^{48} - 1} = 292.955068099 \approx 292.96[/tex]
Therefore, the payment amount made monthly, A ≈ $292.96
2) The total payment made in the 48 months = A × 48 = 292.955068099 × 48 ≈ 14,061.84
The total payment made in the 48 months ≈ $14,061.84
The total interest = (The total payment made in the 48 months) - (The loan amount)
∴ The total interest = $14,061.84 - $12,000 ≈ $2,061.84
39°
.(91-x9)
What solves for x
The greatest common factor(GCF) of LaTeX: x^7,\:x^3,\:x^5\:is\:_{ }__________.
Answer:
x^3Step-by-step explanation:
We are to find the greatest common factor of x^7, x^3 and x^5
x^7 = x^3 * x^4
x^3 = x^3 * 1
x^5 = x^3 * x^2
From both factors, we cam see that x^3 is common to the three, hence the GCF is x^3
the graph of y=-3x+2, state the domain and the range
Answer:
Domain: (-∞,∞)
Range: (-∞, ∞)
Hope this helps.
find the missing measure
Answer:
Step-by-step explanation:
one hundred twenty-three and thirty hundreths. in decimal form
Answer:
0.2330
Step-by-step explanation: