The evaluation of both sides of Stokes' theorem for the given field and rectangular path yields a result of 72 a.
To apply Stokes' theorem, we need to find the curl of the vector field H and then evaluate the line integral around the boundary of the rectangular region in the xy-plane.
First, let's calculate the curl of the vector field H:
curl(H) = (∂Hz/∂y - ∂Hy/∂z)ax + (∂Hx/∂z - ∂Hz/∂x)ay + (∂Hy/∂x - ∂Hx/∂y)az
= 0ax + 0ay + (6x + 6y)az
Therefore, the curl of H is (6x + 6y)az.
Now, let's evaluate the line integral around the boundary of the rectangular region in the xy-plane.
The boundary consists of four line segments:
The line segment from (2, -1, 0) to (5, -1, 0) with positive direction along the x-axis.
The line segment from (5, -1, 0) to (5, 1, 0) with positive direction along the y-axis.
The line segment from (5, 1, 0) to (2, 1, 0) with negative direction along the x-axis.
The line segment from (2, 1, 0) to (2, -1, 0) with negative direction along the y-axis.
Since the positive direction of ds is az, we need to take the cross product of ds with az to get the tangent vector T to the curve. Since ds = dxax + dyay and az = 1az, we have:
T = ds x az = -dyax + dxay
Now, let's evaluate the line integral along each segment:
The line integral along the first segment is:
∫(2,-1,0)^(5,-1,0) H · T ds
= ∫2^5 (6xy)(-1) dx
= -45
The line integral along the second segment is:
∫(5,-1,0)^(5,1,0) H · T ds
= ∫(-1)^1 (-3y^2)(1) dy
= -4
The line integral along the third segment is:
∫(5,1,0)^(2,1,0) H · T ds
= ∫5^2 (6xy)(1) dx
= 81
The line integral along the fourth segment is:
∫(2,1,0)^(2,-1,0) H · T ds
= ∫1^-1 (-3)(-dx)
= 6
Therefore, the total line integral around the boundary is:
∫C H · T ds = -45 - 4 + 81 + 6 = 38
According to Stokes' theorem, the line integral of H around the boundary of the rectangular region is equal to the surface integral of the curl of H over the region:
∬S curl(H) · dS = 38
Since the region is a rectangle in the xy-plane with z = 0, the surface integral simplifies to:
∫2^5 ∫(-1)^1 (6x + 6y) dy dx
= ∫2^5 (12x + 12) dx
= 114
Therefore, we have:
∬S curl(H) · dS = 114
This contradicts the result from applying Stokes' theorem, so there must be an error in our calculations.
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The surface integral of the curl of H over the rectangular region is 0.
Stokes' theorem relates the surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary of the surface. Mathematically, it can be written as:
∫∫(curl H) ⋅ dS = ∫(H ⋅ ds)
where H is the vector field, S is a surface bounded by a curve C with unit normal vector n, and ds and dS represent infinitesimal line and surface elements, respectively.
Given the vector field H = 6xyax − 3y^2ay a/m, we first need to calculate its curl:
curl H = ( ∂Hz/∂y − ∂Hy/∂z ) ax + ( ∂Hx/∂z − ∂Hz/∂x ) ay + ( ∂Hy/∂x − ∂Hx/∂y ) az
= 0 ax + 0 ay + ( 6x − (-6x) ) az
= 12x az
Next, we need to find the boundary curve of the rectangular region given by 2 ≤ x ≤ 5, −1 ≤ y ≤ 1, z = 0. The boundary curve consists of four line segments:
from (2, -1, 0) to (5, -1, 0)from (5, -1, 0) to (5, 1, 0)from (5, 1, 0) to (2, 1, 0)from (2, 1, 0) to (2, -1, 0)Let's calculate the line integral of H along each of these segments. We will take the positive direction of ds to be in the direction of the positive z-axis, which means that for the first and third segments, ds = dxax, and for the second and fourth segments, ds = dyay.
Along the first segment, we have x ranging from 2 to 5 and y = -1, so:
∫(H ⋅ ds) = ∫2^5 (6xy ax − 3y^2 ay) ⋅ dx az = ∫2^5 (-6x) dx az = -45 az
Along the second segment, we have y ranging from -1 to 1 and x = 5, so:
∫(H ⋅ ds) = ∫-1^1 (6xy ax − 3y^2 ay) ⋅ dy ay = 0
Along the third segment, we have x ranging from 5 to 2 and y = 1, so:
∫(H ⋅ ds) = ∫5^2 (6xy ax − 3y^2 ay) ⋅ (-dx) az = ∫2^5 (6x) dx az = 45 az
Along the fourth segment, we have y ranging from 1 to -1 and x = 2, so:
∫(H ⋅ ds) = ∫1^-1 (6xy ax − 3y^2 ay) ⋅ (-dy) ay = 0
Therefore, the line integral of H around the boundary curve is given by:
∫(H ⋅ ds) = -45 az + 45 az = 0
Finally, using Stokes' theorem, we can evaluate the surface integral of the curl of H over the rectangular region:
∫∫(curl H) ⋅ dS = ∫(H ⋅ ds) = 0
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I NEED HELP ASAP!!!!!!!!
Answer: b
Step-by-step explanation:
Teddy is writing a proof about the figure shown on the coordinate plane below. He wants to show that quadrilateral WXYZ is a parallelogram.
Which statement proves that WXYZ is a parallelogram?
A. Show that segments WX and YZ have equal slopes.
B. Show that segments WX and YZ have equal lengths.
C. Show that the midpoints of segments WY and XZ are the same point.
D. Show that segments WX and XY have slopes that are opposite reciprocals of each other.
The statements are -
A. Show that segments WX and YZ have equal slopes.
B. Show that segments WX and YZ have equal lengths.
C. Show that the midpoints of segments WY and XZ are the same point.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is that Teddy is writing a proof about the figure shown on the coordinate plane below. He wants to show that quadrilateral WXYZ is a parallelogram.
The statements that prove WXYZ as a parallelogram are -
A. Show that segments WX and YZ have equal slopes.
B. Show that segments WX and YZ have equal lengths.
C. Show that the midpoints of segments WY and XZ are the same point.
Therefore, the statements are -
A. Show that segments WX and YZ have equal slopes.
B. Show that segments WX and YZ have equal lengths.
C. Show that the midpoints of segments WY and XZ are the same point.
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What is the value of x?
Answer:
x=16
Step-by-step explanation:
according to the geometric mean for altitude
8 squared = 4x
64=4x
x=16
if my answer helped please mark as brainliest.
Step-by-step explanation:
X = 16 .
16 is the correct answer of this question .
plz mark my answer as brainlist if you find it useful.
give answer please.............................fast
Answer:
120
Step-by-step explanation:
Find out the answer for 78^ - 42^.
78^ - 42^ = 4320
36 x = 4320
1 x = 4320 ÷ 36 = 120
x^2y+y^2-30x+221=0 find the radius
Answer:
2
Step-by-step explanation
Let's complete the square. We get:
(x^2-30x)+y^2+221=0
(x^2-30x+225)+y^2+221-225=0
(x-15)^2+y^2-4=0
(x-15)^2+y^2=4
By analyzing this, we see that the radius is just :[tex]\sqrt{4}[/tex]=2
Can someone please help me with this ASAP
Answer:
24.84
Step-by-step explanation:
Hopefully this helps! let me know if it doesn't
HCF of 10,100,1000.
Answer:
So the greatest common factor 10 and 1000 is 10.
plz mark me as brainliest
Think of a favorite dance move, such as the Running Man. Determine whether there is periodicity associated with this dance move.
Answer:
I don't remember if Alan Walker can dance :/
Step-by-step explanation:
Dream did it better than me, you should see his answer.
Answer:
I don't know
Step-by-step explanation:
determine if JK and LM parallel, perpendicular, or neither .
Answer:
3 - Perpendicular.
4 - Parallel.
Step-by-step explanation:
[tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
[tex]JK = \frac{1-(-7)}{-4-(-10)} = \frac{8}{6} = \frac{4}{3}[/tex]
[tex]LM = \frac{-2-2}{-6-(-3)} = \frac{-4}{3}[/tex]
Perpendicular, because two lines are considered perpendicular if their slopes are opposite reciprocals.
[tex]JK = \frac{-2-(-2)}{3-11} = \frac{0}{-8} = 0[/tex]
[tex]LM = \frac{-2-(-7)}{1-1} = \frac{5}{0} = 0[/tex]
BCDE is a rhombus. Find the missing measures.
BC = 28 and BD = 32
CD =
FD =
EF =
EC =
Answer: CD=28
FD= 16
Step-by-step explanation:
someone answer this please!! make sure to consider the function f(x)=3x-1
Answer:
The answers go
2
-1
-4
4
If $30,000 then is equivalent to $900,000;
then, what is $1 then equivalent to in today's
dollars?
Answer:
$30
Step-by-step explanation:
You need to figure out how the money changes between then and now. To do that, take now and divide it by then; 900,000/30,000 = 30, the amount now is 30 times that of then. To figure out what any amount then is worth now, multiply it by 30 and you'll get you answer. (In this case: 1*30 = 30)
I need help answering this question:
-9 ÷ +3
Answer:
-3
Step-by-step explanation:
-9 divided by +3 can be written as -9 divided by 3, because when you have a positive number, you don't have to write a plus sign in front of it. Next, we can just do 9/3 and then add the negative sign to the result to get our answer. 9 divided by 3 is equal to 3, and when you add the negative sign, you get -3.
A right triangle has side lengths d, e, and f as shown below. Use these lengths to find sinx, tanx, and cosx.
Step-by-step explanation:
In the given triangle,
Perpendicular = d
Hypotenuse = e
Base = f
We need to find the values of sinx, tanx and cosx.
We know that,
[tex]\sin \theta=\dfrac{P}{H}\\\\\tan\theta=\dfrac{P}{B}\\\\\cos\theta=\dfrac{B}{H}[/tex],
We have, P = d, H = e and base = f
So,
[tex]\sin \theta=\dfrac{d}{e}\\\\\tan\theta=\dfrac{d}{f}\\\\\cos\theta=\dfrac{f}{e}[/tex]
Hence, this is the required solution.
Using the lengths given, the Trigonometry ratios of the right triangle are:
sin x = d/e cos x = f/etan x = d/fRecall:
In a right triangle where we are given a reference angle, ∅, with three sides, we can solve the triangle using the Trigonometry ratios given as: SOH CAH TOA.Given the following:
∅ = xOpposite = dHypotenuse = eAdjacent = fThus:
sin x = Opp/Hyp = d/e (SOH)
cos x = adj/Hypo = f/e (CAH)
tan x = Opp/Adj = d/f (TOA)
Therefore, using the lengths given, the Trigonometry ratios of the right triangle are:
sin x = d/e cos x = f/etan x = d/fLearn more about Trigonometry ratios on:
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Describe the graph of x > 2 and x < 5. ( i think its a number line for compund inequalitites)
It would have hollow circles on 2 and 5 and you would connect them
can anyone answe this?
Answer:
Corresponding
Step-by-step explanation:
Convert to 0,125km
to centimetres..........
Answer:
12500 cm
Step-by-step explanation:
0,125 km = 125 m (times 1000 because 1 km = 1000 m)
125 m = 12500 cm (times 100 because 1 m = 100 cm)
Bir sayının 2 bölü 5 'sinin 3 eksiği, bu sayının çeyreğine eşittir.
Buna göre bu sayı kaçtır?
Answer:
si cardo bahyanbayansusb
Una pirámide de base cuadrada se corta con un plano paralelo a la base por la mitad de la altura de la pirámide, obteniendo una pirámide más pequeña y un tronco de pirámide. ¿Cuántas veces es más grande el volumen del tronco con respecto al volumen de la pirámide pequeña?
Answer: what does this say?
Step-by-step explanation:
3. Sadie and Josie are asked to wrie an expression equivalent to 3(2x+3y).
Sadie writes the expression 6x+9y.
Josie writes the expression 18xy.
Who is correct and why?
A. Sadie is correct because 3(2x)+3(3y)=6x+9y.
B. Sadie is correct because 3+2x+3+3y=6x+9y.
C. Josie is correct because 3 2x 3y=18xy.
I
D. Josie is correct because 3(6xy)=18xy.
Solve for x to the nearest tenth.
please help i’m failing
Answer:
x≈ 8.3
Step-by-step explanation:
ok first you have to find the hypotenuse of the smaller triangle
a²+b²=c²
2²+4²=c²
4+16=20
√20≈ 4.5
4.5²+7²=c²
20.25+49≈ 69.25
√69.25≈ 8.3
Answer:
x = [tex]\sqrt{69}[/tex]
Step-by-step explanation:
Pythagorean theorem: A^2 + B^2 = C^2
first, find the length of the line cutting through the shape:
let's call this line L.
plug the numbers into the theorem.
4^2 + 2^2 = L^2
16 + 4 = L^2
20 = L^2
next, plug 7 and L into the theorem again to find x
L^2 + 7^2 = x^2
20 + 49 = x^2
69 = x ^2
x = [tex]\sqrt{69}[/tex]
Help me out?!?! tysmmmm
Answer:
1 is isoscele
2 is scalene
3 is equalateral
4 is scalene
Answer for brainliest
Step-by-step explanation:
The diameter is 6, so the square is also 6 feet by 6 feet. This means the square has an area of 36 feet.
Find the area of the circle:
3.14 * 3^2
3 ^ 2 = 9
3.14 * 9 = 28.26
Now find the area of the edges of the square.
36 - 28.26 = ~7.74 ft
PLEASE HELP ME ON THIS ITS THE LAST ONE FOR THE DAY
Answer:
z=95
Step-by-step explanation:
z-28=67
+28 +28
x=95
all you have to do is make both side of the eqaution balanced
if you add 28 to one side you do it to the other side as well
What is the square root of 7.29 ?
Answer:
2.7
Step-by-step explanation:
Answer:
2.7
Step-by-step explanation:
Lana and Andrew gather data by surveying their classmates. Lana asked them about the longest time they had ever run. Andrew asked them about their favorite color.
How could Lana and Andrew represent their data?
Would they represent them in the same way?
Why or why not?
Answer:
No because Andrew aske them being different to Lana an far over.
Which statement describes the congruent triangles
Answer:
it is the first answer
Step-by-step explanation:
Answer:
A
Step-by-step explanation:
you have to line the angles up so the congruent sides lay on each other.
Solve for x in the following figure and choose the appropriate result.
Answer:
2x is equal to 45
e is equal to x and 2
Step-by-step explanation:
Write an equivalent expression by factoring for 6a + 15
pls help with this math question !
thx !
Answer:
h^4
Step-by-step explanation:
Please let me know if you want me to write an explanation for my answer :)