2. Suppose f : X →Y and g : Y → Z are functions, and g of :X → Z is injective. (a) Show that f is injective. (b) Provide an example where f is not injective. (c) Impose a condition on f so that it, together with the assumption that gof is injective, implies that g is injective.

Q: If S is a sphere and F satisfies the hypotheses of Stokes' Theorem, show that curl F· dS = 0.

A: Suppose we have a sphere with upper hemisphere s1 and lower hemisphere s2 Then orientation of…

Q: Evaluate the surface integral. x dS, S is the triangular region with vertices (1, 0, 0), (0, –2, 0),…

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Q: Use Stokes' Theorem to evaluate F• dr where C is oriented counterclockwise as viewed from above.…

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Q: Use the Divergence Theorem to calculate the surface integral F. dS; that is, calculate the flux of F…

A: By the Divergence theorem ∬sF·ds=∭EF·ds The given function is, Fx,y,z=x2z3i^+2xyz3j^+xz4k^ Now…

Q: Use Stokes' Theorem to evaluate curl F· dS. F(x, y, z) = tan-(x²yz²)i + x²yj + x²z?k, S is the cone…

A: The objective here is to evaluate: ∬ScurlF⋅dSWhere, Fx,y,z=tan−1x2yz2i+x2yj+x2z2k; and S is the cone…

Q: Use Stokes' Theorem to evaluate curl F· dS. F(x, y, z) = zei + x cos(y)j + xz sin(y)k, S is the…

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Q: Evaluate the surface integral. x²z² ds S is the part of the cone z² = x2 + y2 that lies between the…

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Q: Evaluate the triple integral. 4x dV, where E is bounded by the paraboloid x = 5y2 + 5z2 and the…

A: We have a paraboloid, and each cross section is a circle, parallel to the plane x = 5. We use…

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

A: Given Fx,y,z=x i+5y j+3z k Surface S is the cube with vertices ±1, ±1, ±1 Using Gauss's divergence…

Q: Use a triple integral to find the volume of the given solid. The solid enclosed by the paraboloids y…

A: Consider y=x2+z2 and y=72-x2-z2      ..... (1) Find the point of intersection…

Q: luate the integral. /2 7 cos?(0) de

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Q: (a) Use Stokes' Theorem to evaluate F. dr where F(x, y, z) = x2yi + x³j + xyk and C is the curve of…

A: Given F→x,y,z=x2yi+13x3j+xykWhere C is the curve of intersection of the hyperbolic paraboloid…

Q: Use Stokes' Theorem to evaluate curl F· dS. F(x, y, z) = x2z2i + y2z²j + xyzk, S is the part of the…

A: Given: F(x,y,z)=x2z2i+y2z2j+xyzk   and S is the part of the paraboloid z=x2+y2 that lies inside the…

Q: Find the center of mass of a solid of constant density bounded below by the paraboloid z = r² + y?…

A: Given, z=x2+y2z=4 Now, r2=x2+y2 Converting the equations for z into cylindrical coordinates,…

Q: Evaluate the surface integral. I. (2z + v²z) ds S is the hemisphere x2 + y2 + z2 = 4, z 2 0 %3D

A: The objective is to evaluate the surface integral,  ∫∫Sx2z+y2zdS where S is the hemisphere…

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

A: Concepts introduction: Let S be an oriented smooth surface that is bounded by a simple, closed,…

Q: Find the average value of the function f(p, p, 0) = p over the solid ball p< 1. Think of f(e, ø, 0)…

A: Given, f(ρ,θ,ϕ)=ρ over the solid ball ρ≤1. To find the average value of the function over the solid…

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: Use Stokes' Theorem to evaluate curl F· dS. Is F(x, y, z) = 2y cos(z) i + eš sin(z) j + xe k, S is…

A: ∬Scurl F·dS=∫CF·dr where C is boundary of the surface oriented counter-clockwise.

Q: The object shown in the figure below is a cube (all edges are equal in length). If the length of…

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Q: 2. Find the volume of the wedge cut from the 1-st octant in R³ by the surface z = 4-1 - y.

A:

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.