Let S be the portion of the solid, D formed by a sphere x²+y²+z² = 16 and xy-plane for which the solid D lies above the xy-plane. Verify Gauss' Divergence Theorem for the following vector field given that S is oriented outward: F = y?zi + 3xzỉ – yz²k

Q: Use the divergence theorem to find the outward flux of the vector field F(x, y, z) = 4x²i + 5y²j +…

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Q: Use the divergence theorem to calculate the flux of the vector field F(x, y, z) = 5xyi + z°j+9yk…

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Q: Find the surface integral F. dS of the vector field F(x, y, z) = (-x, -4, yz®) along the part of the…

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Q: Let S be the portion of the solid, D formed by a sphere x2+y2+z2 = 16 and xy-plane for which the…

A: Gauss Divergence Theorem∬F.ds=∭(∇·F)dv

Q: 3. Use the Divergence Theorem to find the flux of the vector field F = (5x+ 4y?z – 162³) i +…

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Q: Let D be the solid unit cube: 0 < x < 1, 0 < y < 1, 0 < :< 1. Let S be the closed surface boundary…

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Q: Verify that Strokes' Theorem is true for the given vector field F and surface S. F(x, y, z) = yi +…

A: Given : Vector field F and surface S F(x,y,z)=yi+zj+xk Hemisphere ''S'' x2+y2+z2=1,y≥0

Q: Let o be the surface of a cube whose sides have length two. Let o have vertices at the endpoints of…

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Q: Evaluate the surface integral || F. dS for the given vector field F and the oriented surface S. In…

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Q: 3. Verify Stokes' Theorem for the vector field F= 2zi +3xj+5yk, taking S to be a portion of the…

A: Given F=2zi+3xj+5yk

Q: Consider the surface integral of the vector field F = [xy?, 3x², x²z] over a surface S, where S…

A: We will use of gauss divergence theorem

Q: Verify Stokes Theorem for the vector field F(x, y, z) = (y-z)i + 3xy j + 5zk by evaluating the line…

A: note :  As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Let S be the portion of the solid, D formed by a sphere x2+y2+z2 = 16 and xy-plane for which the…

A: Solution is given below

Q: Use the divergence theorem to calculate the flux of the vector field F(x, y, z) = x°i + (5y – 5x*y)j…

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Q: Use the Divergence Theorem to find the outward flux of the vector field F(x, y, z) = 7x3i +…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F=&lt;x,2y,-z&gt; across the…

A: From the Divergence theorem, for a positively oriented boundary surface (outward Normal) of region E…

Q: Let E be the box [0, 4] × [0, 2] x [0, 3] and let S be its boundary. Let F be the vector field F(1,…

A: Let E be the box 0,4×0,2×0,3 and S be the boundary.  Given vector field is  F→=zi→+xj→+yk→…

Q: Let S be the portion of the solid, D formed by a sphere x2+y2+z2 = 16 and xy-plane for which the…

A: please see the next step for solution

Q: |Let F = (x³ – e" )i+(y³+sin(2))j+(2³ – xy)k be a vector field. Use the Divergence Theorem to…

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Q: 3. Verify Stokes' Theorem for the vector field F = 2zi +3xj+5yk , taking S to be portion of the…

A: We will use the basic knowledge of vector calculus to solve this question.

Q: Let F = (x – 2)i + (y – æ)j + (z – y)k be a vector field. Use the Divergence Theorem to evaluate fL,…

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Q: Let F = (x – 2)i + (y – 2)j + (z – y)k be a vector field. Use the Divergence Theorem to evaluate ff,…

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

A: The given problem is to find the flux of the given vector field over the given portion of cone…

Q: Use the divergence theorem to calculate the flux of the vector field F(x, y, z) = x³i + y³j+ z³k put…

A: According to the given conditions

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Q: Verify that the Divergence Theorem is true for the vector field F on the region E. F(x, y, z) =3x i…

A: Given that to   Verify that the Divergence Theorem is true for the vector field F on the region E.…

Q: Use the Divergence Theorem to compute the net outward flux of the vector field F = (x, - y,z") the…

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Q: Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux.…

A: I have solution for 1

Q: Let F = x²yi – xy'j+ (z+2)k be a vector field. Use the Diver- gence Theorem to evaluate ſſ, F ·ndS…

A: In this question, we need to find the flux of vector field by using the Gausses divergence theorem.

Q: Let F = x'yi – xy²j+ (z + 2)k be a vector field. Use the Diver- gence Theorem to evaluate ff,F·ndS…

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Q: Evaluate the surface integral F. ds for the given vector field F and the oriented surface S. In…

A: The S is composed of the surface in the paraboloid in the form y=g(x,z) f(x,y,z)=y-g(x,z)=y-x2-y2…

Q: Let S be the surface in R' that is the part of the ellipsoid 2x2 + y² + z? = 2 satisfying y > 1.…

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Q: Use the extended divergence theorem to compute the total flux of the vector field F(x, y, 2) 3D…

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Q: Use the Divergence Theorem to find the outward flux of the vector field F(x,y, z) = x³i + y³j+z³k…

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Q: Consider the closed, solid region E between the paraboloid z = 4 – x? – y? and the xy-plane, and…

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Q: Evaluate the surface integral F F. ds for the given vector field F and the oriented surface S. In…

A: To find- Evaluate the surface integral ∬SF·dS for the given vector field F and the oriented surface…

Q: Consider the vector field F = (x + y) i+ 2j + zk and the surface S defined as the surface of a cube…

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Q: Use the divergence theorem to calculate the flux of the vector field F(x, y, z) = x³i+y³j+z³k out of…

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Q: Find the surface integral F. dS of the vector field F(x, y, 2) = (-æ, -y, yz®) along the part of the…

A: The complete solutions are given below

Q: Let F (x, y, z) = (y + z2, xy + ez^3, xy) be a vector field, and let S be the closed surface between…

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Q: Use the Divergence Theorem to compute the net outward flux of the vector field F = (x, -y²¸z²)…

A: Given that F=x2,-y2,z2 The objective is to compute the net outward flux of the vector field across…

Q: Let D be the solid unit cube: 0 < x < 1, 0 <y< 1, 0 < : < 1. Let S be the closed surface boundary…

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Q: Consider the vector field F = 2xy i + (x² + 2y) j. Verify the Green's Theorem in the closed region…

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Q: Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux.…

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Q: Find the flux going through just the surface of the cone

A: Consider the provided question,

Q: Use the Divergence Theorem to find the outward flux of the vector field F(x, y, z) = x³i + y³j+ z³k…

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Q: Verify Stokes Theorem for the vector field F(x,y)=(-2yz) i + y j+3x k on the part of the paraboloid…

A: Given F→x,y =-2yzi→+yj→+3x k→  We have to Verify the Stokes theorem for the given vector field on…

Q: diverge

A: answer is in next step

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: