ind the curl and divergence for the given vector field. 1. Positive 2. Negative 3. О div Ĥ 4. O curl H 5. R 6. -k

Q: Given the vector field and oriented curve C, evaluate F.T ds. (x,y) F = on the line segment r(t) =…

A: Given vector field : F=x,yx2+y2 As rt=t,7t; 1≤t≤6 Thus, the parametric representation of x and y…

Q: - draw the vector fields . F(xy) = 2xi – yİk

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Q: Is the vector field F(x, y) = 2xy+y 2xy+ xy+y*x x conservative? Select the correct answer below: O…

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Q: If the vector field v = [mz+x, -y, x]is irrotational, the mis equal to O A. 2 O B. 0 O C. 1 O D.…

A: Curl of v is zero.

Q: What is the value of a so that the vector field F(x,y)=(y eX+4xy3) i+(ax?y-4+ e%) j is conservative?…

A: To find  the value of a for which F is conservative.

Q: Q19. D = yi – xJ, what is the magnitude of the vector field at the point (-1, –1)? -vz 2.

A: 19 Given vector field is V¯=yi¯−xj¯. We have to find the magnitude of the vector field at the point…

Q: (b) Suppose C is a curve of length N, and ||F|| < M, where M is some positive number. Prove that ·…

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Q: nd the curl and divergence for the given vector field. 1. Positive 2. Negative 3. О div H 4. O curl…

A: see below the answer

Q: Match each three-dimensional vector field with the corresponding plot. レ 4レ/ / トー / tト44レ O (x, y,…

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Q: What is the value of a so that the vector field F(x.y)=(In y+2xy³) i+(ax²y2+ x/y) j is conservative?…

A: To find: the value of a for which given vector field is conservative.

Q: Compute the curl of the following vector field. (x, y, z) 5/2 F= (* +y? +z?)* The curl of F is (Di+

A: On solving this problem we will get

Q: Siven the following vector field and orientated curve C, evaluate F•Tds. (x,y) F = on the curve r(t)…

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Q: 10. () The gravitational field F(x) = mMG Ex is a conservative vector field.

A: We shall show that F is conservative vector field by using vector calculus please follow next step…

Q: Jo (b) Let ƒ : IR² → R be a C² function with the property that aº f = 0. Show that the vector field…

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Q: If a vector field F (x, y, z) In (xyz), cos z), which of the following values is equal to div F (1,…

A: Finding divF

Q: Determine the vector field of F. F(x, y) = 6i – 2j

A: We have to identify the vector field for the given vector.

Q: (7) Find (a) the curl and (b)the divergence of the vector field. F (x, y, z) = xye' i_± yze*k

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Q: Q2. (a) Given a vector field, F = 2xyi + (x² + 3y²z)j+ (y³ + 2z)k. (i) Determine whether if F is…

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Q: Which of the following vector field is not conservative? a) Å(r, y) = yi- 3ryj b) 72r + 14(x² +…

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Q: Which one of the following choices is a plot of the vector field given by F = -i+ Va? + y² j? +y²…

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Q: Make a sketch of the following vector field. F= (2y, - x) Choose the correct vector field below. O…

A: The given vector field is F=&lt;2y,-x&gt;

Q: 4) Determine whether the vector field hlxiy)= y?i+ 12xy-e4)T is conservative conservative, deter…

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Q: At how many points is the vector field F(x, y, z) = (z– ,r – 2², y – x²) irrotational? Select one:…

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

A: From the given information.   Observe the given vector field.

Q: Which of the below graphs is of the following vector field? F(x, y) = xi + yj

A: The vector field is a collection of points, where, each point is represented a vector quantity .  A…

Q: What is the value of a so that the vector field F(x.y)=(y eX+4xy³) i+(ax²y4+ eX)j is conservative? O…

A:

Q: 2. Letr =√x² + y² + z². (a) For n ≥ 1 and r> 0, express V(r-n) in terms of r and the radial vector…

A: Given  solution(a): Radial vector field  solution(b):

Q: Consider the vector field v(z) = i+ i+4k . a) Calculate fu• dr , where l' is the straight line from…

A: We’ll answer the first part of this question since due to complexity. Please submit the question…

Q: 4. Assume Á is a constant vector field and R = xi + yj + zk. (a) Simplify the expression (R· VR) ·…

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Q: 4y V16+16g Find the divergence of the vector field F(z, 9) i+ V16z + 16g Provide your answer below

A:

Q: Given the following vector field and orientated curve C, evaluate F-Tds. (x.y) 3/2 on the curve r(t)…

A: Given : vector field F=x, yx2+y23/2, orientable curve C with parametrization           rt=3t2, 8t2 ,…

Q: Q4. A vector field F is defined as F = psin pa,+ 2p² zas + zcos oa,. The V × F at point P(1, 7/2, 2)…

A:

Q: Use the equation giving the flux of the vector field across the curve to calculate the flux of x + 1…

A: The flux of the vector field across is calculated as: Fx,y=x+1x+12+y2,yx+12+y2 or…

Q: Compute the curl of the following vector field. = (6xz° sr. 5xz e 6 ey 6xz° eY) 6. The curl of F is…

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Q: Match the vector field with its graph. F(x, y) = xi + 3yj 山供 得 三 -5+ 5 X 三 -5+ ||| 三 5 BE 州 X x

A:

Q: 3. Find the work of the vector field F(2, y, ) = yi along the curve which is obtained as the…

A: To find the work of the vector field Fx,y,z=yi along the curve obtained by the intersection of…

Q: Compute the curl of the following vector field. F= (5z sin y,5xz cos y, 10xz sin y) The curl of F is…

A: In this question we have to find curl of vector field F

Q: 2 determine the vector evaluated at (2, n/2, n/4). Given the vector field E=rcoso a,+ sine ae +ra

A: Given vector field is The point is From the point:

Q: Which one of the following choices is a plot of the vector field given by P- F: Va2 + y? j? Va2 + y?

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Q: 2 1 Given the vector field F=rcos a.+ =sing a tra evaluate the curl of the vector field.

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Q: What is the value of a so that the vector field F(x.y)=(y eX+2xy4) i+(ax2y3+ e%) j is conservative?…

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Q: What is the value of a so that the vector field F(x.y)-(y eX+2xy4) i+(ax²y3+ e%)j is conservative? O…

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Q: Compute the curl of the following vector field. F = (3xz° ev? 2xz° ey, 3xz? eY*) %3D The curl of F…

A: The given vector field is F=3xz3ey2,2xz3ey2,3xz2ey2.

Q: G is a vector field defined as G = Bxlnzi + Dy²zj + (E+ y³) k. a) Find the values of B and D for…

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Q: For what values of a & b such that the vector field f= (ay'e" + xz)i +(ye" +z')}+| byz' + is…

A: f is given. We find a,b so that f is conservative (irrotational).  f is conservative (irrotational)…

Q: 3. For Parts (a) (b), does the given vector field F appear to be conservative? Clearly answer yes or…

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Q: S [ cos ({*) coslay) de-2sinl4=]sa(Eg]ldy] 1/2,11 251 Show that the vector freld defined under the…

A: 1/2 × Cos(x/2)Cos(2y)dx - 2 × Sin(x/2)Sin(2y)dy  = d( Sin(x/2)Cos(2y) ). Thus, we proved that the…

Q: 5. If F = A × OP, where A is a constant vector field, show that V x F = 2A 6. If R = OP,r = ||R||,…

A: The solution are next step

Q: Compute the curl of the following vector field. F= (x? - y°,5xy,2z) The curl of F is (Oi+ (Di+(k.

A: Topic- vector calculus

Q: Q9. V = 2yī – xJ, what is the magnitude and direction of the vector field made with the 5, 26.6° V5,…

A:

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: