Evaluate the surface integral (double integral) F . dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. Please find for the first octant only. Please show steps in detail. F(x,y,z)=xi+2yj+3zk, S is the cube with vertices (+/-1,+/-1,+/-1))

Q: Find div F and curl F if F(x, y, z) = x²i – 5j + yzk. div F= curl F=

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Q: Calculate grad div F+ curl F at P(-1, 1, 1) where − = (3xz²)i + (xy)j + (2yz)k. O 7i+ 4j - 3k D-3i -…

A: we will find the required.

Q: Find the curl for the vector field at the given point. F(x, y, z)= 6xyzi+6yj+6zk (6, 7, 6) O…

A: To find the curl for the vector field  at the given point.Fx, y, z=6xyzi+6yj+6zkPoint=6, 7, 6

Q: Use (a) parametrization; field (b) Stokes' Theorem to compute fF. dr for the vector F = (x²+z)i +…

A: Given:F=(x2+z)i+(y2+2x)j+(z2-y)kC: intersection of x2+y2+z2=1 and z= x2+y2

Q: How long will it take to drive 700 km at 100 km per hour, if you take a 1-hour break after every 5…

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Q: Find the curl of the vector field F = curl F= [2zy-5xz4] 0 26 Calculator Check Answer X < i + Oj+ x…

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Q: State the theorem for Surface Independence for Curl Vector Fields.

A: Stokes' Theorem Stokes’ theorem says we can calculate the flux of curl F across surface S by knowing…

Q: (b) The square with vertices (0,0), (1, 0), (1, 1) and (0, 1), and the vector field F = 3x²yi + x³j.

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Q: hank you

A: Given,      θ˙=sin2θ

Q: 3) Let S be a portion of the surface 3z = 12 – 6x – 3y in the first octant. Suppose that S is…

A: 3z=12-6x-3y divide by 3 z=4-2x-y ⇒2x+y+z=4 clearly it's plane The plane cuts the x-axis at (2,0,0),…

Q: Let F and G be vector fields with differentiable components. Express curl (F × G) in term of div and…

A: Let F and G be vector fields with differentiable components.Let We will compute .The cross product…

Q: Let F (8xy, 3y, 8z). = The curl of F = (000). Is there a function f such that F = V f? ☐ (y/ (y/n)

A: Step 1: Step 2: Step 3: Step 4:

Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F = (y -…

A: By using green theorem, we calculate the counterclockwise circulation and outward flux of the given…

Q: For the vector field G = (yety + 4 cos(4x + y))i + (xey + cos(4x + y)) 5 I find the line integral of…

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Q: Consider the vector field F(x, Y, z) = 3x?yi+ (x³ + 1) j+9z²k. Let C be the circle cut from the…

A: Given vector field is  F(x,y,z)=3x2yi+(x3+1)j+9z2k C be circle cut from sphere x2+y2+z2=8 by the…

Q: Suppose F 2.xz + 3y²) ay + (4yz²) az. (a) Calculate fLL F dS, where S is the shaded surface in…

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Q: Let FMN be the transformation of the surface. Prove F*dg=d (F*g) gEA (N) = S(N). where

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Q: Find the curl of the vector field at the given point. F (x, y, z) = x²zi - 2xzj + yzk, (2,–1,3)…

A: The solution as follows :-

Q: If F = xyz i+ zxj +xy'z k then find curl F at (1, 2, – 1).

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Q: A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector…

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Q: 8. Use (a) parametrization; (b) Stokes' Theorem to compute fF.dr for the vector field (x² + z)i +…

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Q: Consider the line integral (Image 1) , where C is defined by the vector function: r (t) = - t2i +…

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Q: The circle of radius a centered at the origin, and the vector field F = -i + yj.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: (a) Consider the vector field G = (xyz, yz, xz). Find div(G). (b) (1,2,3). Using the G from part…

A: We are given the vector field  G=xyz,yz,xzwe have to find the divG , line integral ∫cG.dr for line…

Q: a) Find the values of a,b,c which make f(x,y,z) = sin (xªy°z©) a potential function for the vector…

A: I have solved first part of your query. I request you to repost your query for next part

Q: (b) Use Green's theorem to calculate the circulation of F around the triangle with verticcs (2,0),…

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Q: Find curl(zi + xj+ yk).

A: Using definition of curl of vector.

Q: 9. Find the flux of the vector field B(x, y,=) shown in the left figure through the square S of - yỉ…

A: The general form of a vector valued function in three dimensions is given by…

Q: 12. Evaluate the curl of the following vector field at the point (0,1,2): F = (x + y + z, x? + y? +…

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Q: Find div F and curl F if F(x, y, z) = x*i – 11j + yzk. div F= curl F=

A: Vector

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: