Please solve the screenshot! (Example is the other screenshot.) EXAMPLEIf F(x, y) = (-yi+ x j)/(x² + y²), show that f. F· dr = 2n for everypositively oriented simple closed path that encloses the origin.SOLUTION Since Cis an arbitrary closed path that encloses the origin, it's difficult tocompute the given integral directly. So let's consider a counterclockwise-oriented circle C'with center the origin and radius a, where a is chosen to be small enough that C' liesinside C. (See Figure 1.) Let D be the region bounded by C and C'. Then its positivelyoriented boundary is C U (-C') and so the general version of Green's Theorem givesyC'P dx + Q dy + P dx + Q dy = ||õeӘРdAдуD-C'y? – x?(x² + y²)²y? – x?(x² + y²)².dA = 0FIGURE 1ThereforeРӑх + Q dy 3D | Pӑх + QdyS. F • dr - F• drthat is,We now easily compute this last integral using the parametrization given byr(t) = a cos ti + a sin t j, 0 < t < 27. ThusF· dr = F· dr = " F(r(t)) · r(t) dtJc'(27 (-a sin t)(-a sin t) + (a cos t)(a cos t)a cos?t + a²sin?tdt =" = 27 If F is the vector field of this example, show thatF. dr = 0 for every simple closed path that does not pass through or enclose the origin.

Given the vector field F(x,y)=P,Q=-yx2+y2,xx2+y2. And from the example above, we have ∂Q∂x-∂P∂y=0…

Answered: If F is the vector field of this…

If F is the vector field of this example, show that F. dr = 0 for every simple closed path that does not pass through or enclose the origin.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 5CM: Take this test to review the material in Chapters 4 and 5. After you are finished, check your work...

See similar textbooks

Related questions

Q: Find a formula for the vector field (vectors are shown as line segments, with a dot at the tail of…

A: This question is about application of derivative

Q: The vector field F is shown in the xy plane. 3- 2- 2 3 4 Is div F positive, negative, or zero?…

A: Divergence is a operator which operate on vector field F to gives a scalar quantity . And divF is…

Q: Give an example of a nonzero vector field F such that curl(F) = 0 and div(F) = 0.

Q: Show that if a vector field has the form F = ⟨ƒ(y, z), g(x, z), h(x, y)⟩, thendiv F = 0.

Q: Find a vector field with twice-di£erentiable components whosecurl is x i + yj + z k or prove that no…

A: To find: The vector field with twice differentiable components whose curl is xi+yj+zk, prove that no…

Q: Consider the vector field F that is depicted below. On which of the parameterized curves is the…

A: Given the vector field F in the image

Q: Show that the vector field F = ( -z, 0, x) is orthogonal to the position ----+ vector OP at each…

A: Given data: The vector field in the Cartesian co-ordinates is given by, F=-z i+x k. Here, I and k…

Q: Let V and V' be n and m dimensional vector spaces over a field FE T

Q: If V is a vector space of dimension n over the field Zp, how manyelements are in V?

Q: Make a sketch of the following vector field. F = (-y,x) ..... Choose the correct vector field below.…

A: We have to match a sketch of the vector field F = < -y, x>

Q: Find the curl of the vector field F = curl F -

Q: Determine and state the work done by the vector field over the given path in each situation. F…

Q: Find the divergence of the vector field F =

Q: -2t y= c1e -1 1

A: For the solution of the problem follow the next step.

Q: Find the divergence of the vector field F = div F

Q: For any vector field F, show that div(curl F) = 0.

A: We need to prove that for any vector field F, div(curl F) = 0.

Q: Sketch the following vector fields:

A: given vector field <cosx2+y2 , sinx2+y2> first component is cosx2+y2and second component…

Q: Write a formula for a two-dimensional vector field which has all vectors of length 7 and…

Q: Show that F = (3, 1, 2) is conservative. Then prove more generally that any constant vector field F…

A: Given: F = (3, 1, 2) To show: F is conservative. Also to show any constant vector field F = (a, b,…

Q: What are the outputs of a vector field in ℝ3?

A: What are the outputs of a vector field in ℝ3.

Q: Consider the vector field F = ( 'x²+y2 'x²+y2/ Write yes or no for each: Is this field conservative…

A: The solution are next step

Q: Given two circles centered at the origin, oriented counterclockwise, and any vector field F, then…

A: Line integral is integral of some function along a curve or path. Any field for which the line…

Q: If F is a vector field and there exist a closed curve C such that SF. dr = 0 then Fis conservative.

A: Since you have posted a multiple question according to guildlines I will solve first question for…

Q: Show that if a vector field has the form F = ⟨ ƒ(x), g(y), h(z)⟩, then∇ x F = 0.

Q: Show that V² (v-x) = 2V.v+x-V²v, where v is a vector field and x is the position vector of a point…

Q: Determine whether the line integral of each vector field (in blue) along the oriented path (in red)…

A: Given below clear explanation

Q: Sketch the vector field F = (5x.4y - 5x). Choose the correct answer below. O A. OB.

A: Given query is to find the correct option for the vector field of F.

Q: let c is a differentiable vector field then div(curl)=

A: This question is related to vectors.

Q: State whether the following statement is true or false in a vector space y over a field F: uEV such…

A: We assume that the vector u is non zero.

Q: Which vector field F has graph y

A: Given: Vector field graph is given

Q: Write a formula for a two-dimensional vector field which has all vectors parallel to the y-axis and…

Q: Given that F and G are vector fields with G a vector potential of F, prove that G is not unique

A: We will simply use the definition of "vector potential" of a vector field function.

Q: 2. Find the divergence of the vector field F div F = %3D

Q: 5. Show that there is no vector field G such that: V x G = 2xi + 3yzj – xz?k.

A: Given that ∇×G→=2xi^+3yzj^−xz2k^. We have to show that there is no vector field G→ such that…

Q: (a) Define a vector field in 2-space with an F(x, y) = yi + xj example. Graph of the vector field

A: For example magnetic field of a bar magnet. The field lines can be visualized using small iron…

Q: Find a vector field with twice-differentiable components whose curl is x i + yj + z k or prove that…

Q: Find the divergence of the vector field F .

Q: i) Show that F is a conservative vector field.

Q: Determine whether the following vector field is conservative. If it is , find the potential…

A: F=<cos(z0, 2y-xsin9z)>=cosz i+2yi-∅sinz k i) curl F=ijk∂∂x∂∂y∂∂zcos z 2y-xsinz…

Q: Define the following :- i) Vector Field. #) Piecewise Smooth Curve C

Q: For every vector field F on R³, curl(divF) = 0

A: Explanation of the answer is as follows

Q: Select the option that gives the vector field that could correspond to this arrow map.

A: The vector field is when a vector is assigned to each point in the given space. The cartesian form…

Q: Sketch the following vector fields: 1.

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Given two circles centered at the origin, oriented counterclockwise, and any vector field F, then…

Q: If the scalar field is given by f = 2p², the result of V × Vƒ at P(4, ,4) 3

A: The expression for the gradient in the cylindrical coordinates is given as . Following are the dot…

Q: Determine whether the following vector field is conservative on R. F- (x.4y)

Q: Give an example of a non-zero vector field that is not irrotational and not incompressible.

A: We find an example of a non-zero vector field that is not irrotational and not incompressible.

Concept explainers

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

100%

Please solve the screenshot! (Example is the other screenshot.)

EXAMPLE
If F(x, y) = (-yi+ x j)/(x² + y²), show that f. F· dr = 2n for every
positively oriented simple closed path that encloses the origin.
SOLUTION Since Cis an arbitrary closed path that encloses the origin, it's difficult to
compute the given integral directly. So let's consider a counterclockwise-oriented circle C'
with center the origin and radius a, where a is chosen to be small enough that C' lies
inside C. (See Figure 1.) Let D be the region bounded by C and C'. Then its positively
oriented boundary is C U (-C') and so the general version of Green's Theorem gives
y
C'
P dx + Q dy + P dx + Q dy = ||
õe
ӘР
dA
ду
D
-C'
y? – x?
(x² + y²)²
y? – x?
(x² + y²)².
dA = 0
FIGURE 1
Therefore
Рӑх + Q dy 3D | Pӑх + Qdy
S. F • dr - F• dr
that is,
We now easily compute this last integral using the parametrization given by
r(t) = a cos ti + a sin t j, 0 < t < 27. Thus
F· dr = F· dr = " F(r(t)) · r(t) dt
Jc'
(27 (-a sin t)(-a sin t) + (a cos t)(a cos t)
a cos?t + a²sin?t
dt =
" = 27

If F is the vector field of this example, show that
F. dr = 0 for every simple closed path that does not pass through or enclose the origin.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Linear Equations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.