EXERCISE 8: GAUSS’S THEOREM (Answer all) 1. Given vector field F = 4xi +3y j+zk,where surfaces S are the closed surfaces of the solid region bounded by 0

Q: Let S be the part of paraboloid z = 9 – x2 - y? with z 2 0. Verify Stokes' theorem for vector field…

A:

Q: I Verity Green's F (8,y) = Cx-xy) and the part D={Cs,x) F": theorem for the vector fiela

A: Introduction: Green's theorem connects a line integral around a simple closed curve C to a double…

Q: Draw the unit cube D = {(x, y, z2: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1} and sketchthe vector field F =…

A: Given:The unit cube Vector field Objective:Sketch the vector field  on the six faces of the…

Q: Calculate div F and curl F.

A: F(x,y) = F1i + F2j F1 = x - y2 + 2 F2 = xy + y

Q: Let S be the part of paraboloid z = 9 - x² - y² with z > 0. Verify Stokes' theorem for vector field…

A: Let S be the part of paraboloid z=9-x2-y2 and vector field Fx , y , z=4zi^+5xj^+2yk^. To Verify the…

Q: If (a, b) is a local maximum of f, then D, f(a, b) = 0 for any unit vector u e R?. True False

A: Question: if (a,b) is a local maximum of f, then Du→f(a,b)=0 for any unit vector u→∈R2.

Q: F(x, y, z) = (x²)Î + (2y³2)ĵ + (32)Â is a vector field. (a) div(F) (b) curl(F) (c) C is the…

A:

Q: Verify Green's Theorem in the plane for F = xyî + x²ĵ and where C is the boundary of he region…

A:

Q: let F be the vector field Cx, 2x,3). let c be the of the Surface S gien by 4xt y1855 it. 871.Given c…

A: We will use reverse of Stokes theorem for vector  field F to convert given integral into line…

Q: -1 15. Determine if span R2. 1

A: A set of vector S is said to be span Rn if every vector of Rn can be written as the linear…

Q: Calculate sin(x) -dx. √cos(x)

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

Q: Consider the vector field F = p? + yzi+ + In(s* + :² + 1)k „j + In(=* + :² + 1)k 12 + y²" defined on…

A: We will use the basic knowledge of vector calculus and integral calculus to answer this question.

Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: Let S be the square in the zy-plane shown in the figure below, oriented with the normal pointing in…

A:

Q: In the puzzles below each letter represents distinct digit in a decimal scale. Identify the digits.…

A: Given  7.FLYPIG=6.PIGFLY

Q: a) Show that F is conservative while G is not. b) Evaluate GdR, where C3 is defined by R(t) = tî +…

A: This is a problem of vector calculus.

Q: Let vector field F =(x,y,z²) across the surface S which is an upper hemisphere x²+ y² + z² = 4 . 1)…

A: Let vector field F =(x,y,z²) across the surface S which is an upper hemisphere x²+ y² + z² = 4 . 1)…

Q: b) Given the vector field OU F(x, y, z) = 3z cos x i+ x ln y j +e° tan a k. Determine i. div F, ii.…

A:

Q: If a vector field F (x, y, z) = (y² + z², xe cos z, - xe³ cos z), which of the following values is…

A: Given vector field is Fx, y, z=<y2+z2, xey cosz, -xey cosz>.   To Find: div F.

Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

A:

Q: Find the area of the portion of the unit sphere inside the cylinder x2 + y? = 1/2 and z > 0. Select…

A:

Q: Determine whether F(x, y, z) = 2zi + (3y-y²)j + (2x − 23k is a conservative vector field. → We are…

A:

Q: A vector field is given as F = âr2r“ + a¿r²CoSo + âz5z. Verify Stokes theorem for this vector field…

A:

Q: Given the vector field F = (r + y)7+ (r – y)7, evaluate it's line integral over the plane curve a? +…

A: Line integral

Q: Consider the vector field F(x, y, z) = (xe² ye²³, -2e²) Let S be the surface defined by {X²+y²…

A: Answer :-

Q: Seven office mates, three women (Fran, Gladys and Helen) and four men (John, Karl, Leonard and…

A: ii. 4Explanation:Restrictions:i. Fran, the only day she eats at a restaurant is Wednesday and on the…

Q: 26. Verify Stokes' Theorem for the vector field F = 2xyi + xj + (y+z)k and surface z = 4x² - y², z ≥…

A: The given problem is to verify the stokes theorem for the given vector field over the given surface…

Q: a) Show that if u, v, and w are differentiable vector functions of t, then d du V X w +u - dt dv dw…

A: According to our guideline we are supposed to answer only one asked question.kindly repost other…

Q: 1: Let Let F = (yz², tan-1(xy), 1 + xy³). %3D a) Show that F is not conservative. b) Show that F…

A: conservative vector field is a vector field that is the gradient of function

Q: AK5: Let S be the part of the graph of the plane z = 1- x - y/2 over the triangle 0≤ y ≤2-2x and 0 ≤…

A: Stokes theorem: If S is a surface enclosed by a closed curve C, then

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: