5. Consider the graph of the function f(x, y) = 4 – (x² + y²) over the region R: x² + y² < 4. (a) you may not want to integrate in Cartesian). Find the surface area of the portion of the graph of f that lies over the region R (Hint: Find the flux of the vector field F(x, y, z) = xi+yj over the portion of the graph that (b) lies over the region R.

Q: Verify that f (x, y, z) = xy + yz 2 is a potential function for the vector field F = (y, x + z2,…

A: Given: Vector field F→=y , x+z2, 2yz To verify that fx , y, z=xy+zy2 is the potential function.

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

A:

Q: 4. Find the flux of F across S if F = yi-xj+2zk and S is the hemisphere x² + y² + z² = 4, z ≥ 0,…

A: We have to find flux passing through sphere.

Q: Compute the flux of F = xi+yj+zk through the curved surface of the cylinder x² + y² = 1 bounded…

A: The given vector field is F→=xi+yj+zk. The goal is to evaluate the flux or surface integral of the…

Q: Show that F : R^2 → R^2,F(x,y) = (x + y^2,2xy + 2y^2) is a gradient vector field, and find the…

A:

Q: Let divF(x,y, z) = 4x³ + 4y³ + 42³ , where F represents the flow of a vector field. Estimate the…

A: Concept Used:- Divergence Theorem Flux=∫∫∫∇.F dVwhere ∇.F is the divergence of FVolume of the…

Q: 8. Find the flux of F = (2x – 5y², 3y, x) out of the sphcre of radius 2 centered at the origin.

A: The flux of the vector space is given by the formula, ∫S∇.F dS where S is the sphere of radius 2…

Q: The outward flux of the vector field F(x, y, z) = (x, y, z') across the boundary of D- {(x, y. z)…

A: Using divergence theorem calculate the flux:  φ=∯SF.dS   =∰V∇.F.dv…

Q: Compute the flux of the integral of the vector field F(z,y, 2) = (2,y, 2) through the half cylinder…

A: Given vector field is, Fx,y,z=x,y,z. It can be written as,  Fx,y,z=xi^+yj^+zk^. Flux of vector field…

Q: II. Let S be the portion of the cylinder x² - 6x + z = 4, oriented by upward-slanting normal…

A:

Q: Find the flux of the vector field F(x, y, z) = [xy, yz, zx] through the cylinder x2+y2 <= 1, 0 <= z…

A: The given problem is to find the flux of the vector field through the given surface of the cylinder…

Q: nt) Compute the flux of the vector field F = 5xi + 5yj through the surface S, which is the part of…

A:

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: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = e F(x, y, z) = ( 9z+5y+6z

A:

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z)=tan(4x+y+z). F(x, y, z) -

A: Solution:-                The given function is f(x,y,z) = tan(4x + y + z) then we have to find…

Q: calculate the flow of the vector field a(M) = (x-y)i + (x+y)j +(z2)k through the surface of a…

A: aM=x-yi^+x+yj^+z2k^ The equation of the cylinder x2+y2=1, z=0, z=2 The flow is in the direction of…

Q: Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 5 for fluid flowing…

A:

Q: The gradient vector for the surface xy + xz + yz = 1 at the point (1, 1,0) is (1, 1,0). Select one:…

A:

Q: Find the flux of the field F(x, y, z) = (x, y, -2z) through the surface z = ! =1– 2² – y² (0 < ¤< 1,…

A:

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: Compute the flux of the vector fields F(x,y, z) = (, y, z) across the 1 and z = 4 whose vertical…

A:

Q: Let divF(x, y, z) = 4x³ + 4y3 +4z³, where F represents the flow of a vector field. Estimate the flux…

A: Given divF→x,y,z=4x3+4y3+4z3radius r=0.1 Centered at point 1,1,1

Q: F (x,y,z) =(2x+ cos(yz) )i +(xe^-z -y)j+ zk vector field D: x^2 + y^2 =4  ,  x+z=2  , z=0…

A: Given vector field is, Fx,y,z=2x+cosyzi+(xe-z-y)j+zk.

Q: F (x,y,z) =(2x+ cos(yz) )i +(xe^-z -y)j+ zk vector field D: x^2 + y^2 =4  ,  x+z=2  , z=0…

A: Given : vector field F→(x,y,z)=(2x+cos(yz))i→+(xe-z-y)j→+zk→D: x2+y2=4,  x+z=2,  z=0To find :…

Q: Find the total upward flux of the field F = across the surface σ: z =1 – 4x^2 – 4y^2 above the…

A:

Q: (c) Calculate the flux of the vector field F = xi+yj + zk outwards the curved surface of the…

A:

Q: Use Stokes' Theorem to calculate the line integral of the vector function F = (y, z, x) along the…

A: Given vector function: F=y,z,x The Stokes' theorem is given as: ∫CF→·dr→=∬R∇×F→·n^ds Curl…

Q: Let divF(x, y, z) 4x3 + 4y3 + 4z³ , where F represents the flow of a vector field. Estimate the flux…

A: Given that a small sphere of radius 0.1, centered at the point  (1, 1, 1).And                  div F…

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: