#7: Consider the following vector field. F(x, y, z) = 3yz In xi + (9x – 6yz)j + xy°z³ k %3D (a) Find the curl of F evaluated at the point (8, 1, 4). (b) Find the divergence of F evaluated at the point (8, 1, 4).

Q: Q2. If F = (x² - y² + 2xz)ī− (2xz − xy + yz)j + (z² + x²), find the divergence of curl F and curl of…

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Q: Q4: Find the curl of the vector field F(x, y,2) = xJyi +j+ zk at the point(2,4).

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Q: Consider the vector field F(x, y) = (3y² – 2xy, 6ry – x²). (a) Show that F is conservative. (b) Find…

A: According to question, Given vector field, F→x, y = 3y2 -2xy, 6xy-x2

Q: Consider the given vector field. F(x, y, z) = 7e*Y sin(z)j + 6y tan-'(x/z)k (a) Find the curl of the…

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Q: The following figure shows a half-cylindrical closed surface, defined by the cylinder x² + y² = 4…

A: As per the question we have to find the total flux of the vector field V through the flat top…

Q: Consider the vector field Ĥ(x, y, z) = (y², x, z²). Let S3 be the portion of the paraboloid 2 = x² +…

A: Please comment if you need any clarification. if you find my answer useful please put thumbs up.…

Q: Q4 A) Calculate the divergence of the vector field F=y³i + xyj B) Find the Curl of F(xy.z)=3x²i +…

A: To find out the divergence of vector field.

Q: Consider the gradient vector field V = Vg of the function g(x, y) = x² + y² + e*²-y´ cos(2xy).…

A: We will use the Divergence Theorem in 2D to solve the given flux integral.

Q: Use the Divergence Theorem to compute the net outward flux of the vector field F across the boundary…

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Q: e) Find the divergence of the vector field at the given point: F (r, y, 2) = In (æyz) (i +j + k);…

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Q: Let + y³)%¸ æyz 7 4 F = x² + be a vector field on U = {(x, y, z) E R° | (x, y) + (0, 0)}. Find its…

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Q: 3. Consider the vector field F = (y, -x, x² + y?). Find the flux of F through S if: 9– x2 – y? above…

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Q: (a)the curl and (b)the divergence of the vector field. F(x,y,z)= ln(2y+3z)i+ln(x+3z)j+ln(x+2y)k

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Q: Find the divergence of the vector field F = < ya®, æ2®, zy* div F = Preview

A: Divergence of F→=yx3, xz6, zy4

Q: Use the divergence theorem to calculate the flux of the vector field F(x, y, a) + (2y-3')] + 7k…

A: Solution :

Q: Consider the vector-field F(x, y) = y²i-xj, and let C be the arc of the parabola y = k – x² from the…

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Q: 34. F = x³i + y*3 + (x² + y²) k and S is the closed cylinder x? + y² = 4 with 0 < z < 5.

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Q: b) Find the flux of the vector field F(x,y,z) = 3xi – yj + 2zk; across o where o is the portion of…

A: The given question belonging to the vector calculus  in which we have to find the value flux of the…

Q: Find the curl and divergence of the vector field. F(x,y,z)=x^2yz i -x x^3y k

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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: Let W be the work done by the forced field F(x,y,z) =yi+zj in order to move a particle along the…

A: The formula for work done is W=∫F.dr.

Q: Find the divergence of the vector field F→(x,y,z)=7sinxi→−cos(3y)j→+z^9k→.   Enter the exact…

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Q: Determine the divergence of these vector fields: (a) P = xʻyz a, + xz a, (b) Q3D p sin фа, + p'za,…

A: The given problem is to find the divergence of the given vector fields, We have to use Cartesian ,…

Q: Find the flux of the vector field F across the surface S in the indicated direction. F =x3y 2i+yj…

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Q: Q1) Determine the gradient of the following scalar field V = e(2x+3y)cos5z Q2) Given f = x²y + 3z +…

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

Q: Find the divergence of the vector field F at the given point. F(x, y, z) = xyzi + xz²j + 3yz²k; (2,…

A: This question is based on divergence of vector.

Q: find the work done by the given vector field F along the specified curve C.   F(x, y) = yi − x j,…

A: Given, F(x,y)=yi^-xj^ where curve, x2+y2=4 W=∫F.dr W=∫ydx-xdy from curve, dx=-dy W=∫xdx+ydy limit of…

Q: Let ryz , (교2 + y"), 17 F = ( x2 +y? be a vector field on U = {(x, y, z) E R³ | (x, y) # (0, 0)}.…

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Q: Given the vector field F(x, y, z) =(z cos(xz) + ze“ )î +(3y² +1)j+(xcos(xz) + xe )k 4. (a) Find the…

A: Given F→x,y,z=z cosxz+zexzi^+3y2+1j^+x cosxz+xexzk^

Q: Consider the vector field F (x, y, z) = (ycosh x + ze*, sinh x – 3z3, e* – 9yz²) %3D a. Find the…

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Q: 1. Consider the function f (x, y, z) = y- + y² +z? Determine the gradient vector field of f, F =Vf…

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Q: Ex: Find the work done by the vector field F = (y-x²)i + (z - y²)j + (x-z²2)k Over the curve r(t) =…

A: Given: F=y-x2i+z-y2j+x-z2krt=ti+t2j+t3k

Q: Q6: Find the outward flux of the vector field F=(x²y+e'sin(y))i+ (x²+ e*cos(y))j through the right…

A: Solution

Q: For Problem 2, calculate the flux of the vector field F through the given surface S. 2. F = -4i+…

A: Given F→=-4i^+xj^+2zk^ and S:x+2y+z-2=0

Q: 9) Find the (i) divergence and (ii) curl of the vector field F(2, y, 2) = (4yz) i+ (y²z) j + (ryz) k

A: We have to find curl

Q: Use the divergence of the vector field to compute F(x, y) = (tan-1 (2²)) i + (xy-2 +82²) j 4 fo…

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Q: Find (a) the divergence and (b) the curl of the vector field F(x, y, z) = cos(z)i + 5ysin(5z)j+…

A: Given vector field is  F(x,y,z)=(cosz)i+5ysin(5z)j+2x2zk

Q: 2.4) Use the divergence theorem to find the outward flux of the vector field F(x, y, x) = x³i+ y³j+…

A: The divergence of the given vector field is  ∇·F=∂∂xx3+∂∂yy3+∂∂zz3=3x2+3y2+3z2 Let, S be the surface…

Q: Find the divergence of the following vector field. (x,y,z) F= 9+ 3x? +y? The divergence of F is

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Q: Find the divergence of the vector field F(x, y, z) = In(x² + y? )i+xyj+ln(y² + 2²)k. 2x 2z +x + x2 +…

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Q: Find the divergence of the vector field F. F(x,y,2) %3D 6хe\i + 4ye'j

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Q: 21. Compute the flow of the vector field F = r through the surface of the ellipsoid y² + 2 = 1 in…

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Q: Answer two of the following (A) Suppose C= 4x²y° + i3x³y² . Find divergence of C. (B) If A = x³ y z°…

A: As per the answering guidelines we need to answer only one question so please repost the other one.…

Q: Find the counterclockwise circulation and outward flux of the field F = 7xyi + -5y²j C enclosed by…

A: Introduction: Let C be a piecewise smooth, simple closed curve enclosing a region R in the plane.…

Q: Find the divergence of the vector field V(x, y, z) = -2e*'i – e³xyj+ 7e%yzk.

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Q: - Here is a graph of the vector field F(r, y) = r²i+ y°j. 2 (a) Calculate div F. (b) For which…

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Q: 3. What is true about the value of F.dS for the vector field F (r'y,-ry,-I*yz) where S is the…

A: Solution 3: Given  F=x3y,-x2y2,-x2yz We have to calculate ∬SF.dS, where S is the surface of…

Q: Consider the following vector field. F(x, y, z) = (2e* sin(y), 7e sin(z), 4e² sin(x) (a) Find the…

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Q: Consider the one-form w = x' cos(y) dx + xe13% dy+4z² dz on R3. The associated vector field is: F(x,…

A: For the solution follow the next step.

Q: Evaluate F Î ds for the vector field F(x, y) = yì – xj counterclockwise along the circle x² + y² = 1…

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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: