2.4) Use the divergence theorem to find the outward flux of the vector field F(x, y, x) = x³i+ y³j+ z°k V1- x2 – y? and accross the surface of the region that is enclosed by hemisphere z = che plane z = 0. |

Q: Find the flux of the vector field F = (y, z, x) across the - part of the plane z = 4+3x+2y above the…

A:

Q: Find an equation of the plane through the origin such that the circulation of the field F = zi + xj…

A: Given the circulation of the field F = zi + xj + yk along the intersection of the planeand the…

Q: a) Compute the surface area of the portion of the plane x+y=z = 0 contained inside the cylinder x² +…

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Q: Find the flux of the vector field = (y, −z, x) across the part of the plane z = 1 + 4x + 2y above…

A: Step 1:Step 2:Step 3: Step 4:

Q: 4. Evaluate the flux of vector field F = zi + 2yj - 2xk across part of the paraboloid z = 4-x² - y²…

A: A vector field F=zi^+2yj^-2xk^ across part of the paraboloid z=4-x2-y2 that lies above the circle…

Q: Consider the force field F(x, y) = (x, y). a) Sketch the vectors F(1,1), F(-2,1) and F(2,0) directly…

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Q: Find the flux of the vector field F across the surface S in the indicated direction. F = x 2y i  -…

A: To find the flux of the vector field F across the surface S. F=x2yi−zk S is the portion of the cone…

Q: 2. Use Gauss' Theorem to compute the outward flux of the vector field F(r) = (x, y, z²) through the…

A: Outward flux by using gauss divergence theorem

Q: Compute the flux of the vector field F =2xi +3yj through the cylindrical surface shown below, if a=3…

A: Given that the vector field is F(x,y)=2xi+3yj Let us parametrize the cylinder as x=cosθ, y=sinθ, z=r…

Q: Sketch (a) the level surfaces of the scalar point function f(r) = ze; (b) the field lines of the…

A: Given Information:a) Scalar point function: f(r)=ze−xyb) Vector point function: F→(r)=(−y,x,1)To…

Q: Compute the flux of the vector field F(x, y, z) = (2,0, x³) across the portion of the plane x-z = 0,…

A:

Q: a) Find the equation of line of intersection of the following planes 3x – 4y + 4z - 29 = 0 , 6x – 8y…

A: As per guidelines we are allowed to solve one question at a time. Kindly repost the rest question…

Q: Find the flux S S;F · dS of the constant vector field F(x, y, z) = (0,0, 1) across the surface S…

A: Let F be a constant vector field over a surface S given as  F(x,y,z)=0,0,1  The surface S is…

Q: Find the flux of the vector field F(x, y, z) =(x)i+(y)j+(z)k through the surface S of the…

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Q: The velocity vector field for a fluid flowing in R3 is given by = (xz, -yz, y) m/s. Let S be the…

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Q: Use the divergence theorem to calculate the flux of the vector field F(x,y,z)=−3xy−2yzj+4xzk…

A: Given: And the radius of sphere S is 3.To Do: Calculate the Flux using the divergence theorem.

Q: Consider the curve C from (-5, 0, 1) to (6, 4, 3) and the conservative vector field F(x, y, z) = (yz…

A: Answer :-  Your answer is correct but don't write fractional  part  because answer is in whole part…

Q: Calculate the divergence and the curl of this vector function.

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

Q: Find the flux of the vector field F = xi + ej z k through the surface S given by that portion of the…

A:

Q: Find the flux of the vector field (y, x, z) in the negative z direction through the part of the…

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: