d: R? x R? R Q2: Let d(a, b) = |b, - al + do(az, b2), where do denotes the discrete metric on R Prove that d is a metric on R?. a = (a,,a2); b (b, ,b2) E R?, define the function by

Q: Use Stoke's Theorem to evaluate Į F· dr where F (x, y, =) = , and C is the triangle with C vertices…

A: given Fx,y,z=z2,y2,xy ⇒Fx,y,z=z2i^+y2j^+xyk^ and a traingle C with vertices 1,0,0 ,0,1,0 ,0,0,2 to…

Q: Evaluate f. F.dr, where F(r, y, z) = z?i+rj and C is the boundary of the square {(r, y, z)|0 < x <…

A:

Q: Let x = x(u, v) be a regular parametrized surface. A parallel surface to x is a parametrized surface…

A: Let x=x(u,v) be a regular parametrized surface,  A  parallel surface to x is a parametrized surface…

Q: Let d: R? × R² → R, defined as d (æ, y) = |æ1| where a = ("1, #2), y = (y1, 42) , then d is metric…

A:

Q: Let r(t) be a positively oriented rectangle with corners (0,0), (0,1), (2,1) and (2,0) and F(r, y)…

A:

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

A: The given vector field and region C is as follows.

Q: Let P and Q be points in R2. Show that a linear map G(u, v) = (Au+ Cv, Bu+ Dv) maps the segment…

A: Let P and Q be two points in R2. We have to show that a linear map G(u, v) = (Au + Cv, Bu + Dv) maps…

Q: Determine whether the space curve given by r(t) = (t, t 3 , t 2 + 1) intersects the xy-plane, and if…

A: Given the space curve,  r(t)=t,t3,t2+1

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

A:

Q: Let (x, y) and (z, w) be two points in R? state which of the following does not define metric on R?…

A:

Q: Evaluate the curvature of r(t) at the point t = 9. %3D r(t) = (3 – 21, est45, 18 – 1²) (Use symbolic…

A:

Q: Find the Curvature of the space curve r (t) = t i + t? j + -2 t3 k at t = Enter an integer or a…

A:

Q: Consider the following mappings on RXR: * d1(1, y) = e²=y, d2 = |2r - 2y). and d3 = V12r – yl. %3D…

A:

Q: Find T, N, and k for the space curves r(t) = (3sint)i + (3cost)j+ 4tk %3D

A: As per our guideline we are supposed to answer only one question you can repost other questions in…

Q: Determine whether the statement is true or false. If D is the disk given by x2 + y2 s 49, then 49 x2…

A:

Q: For the trivial line bundle E over the manifold M = R onsider the connection 1-form w =-xzdt + (x² –…

A: Given: Here, the given is for the modified bundle E over the trivial bundle E M=ℝ4. Consider the…

Q: The figure shows a parallelepiped spanned by vectors a,b and c Which of the following formulas…

A: The formula to calculate the volume is given by, Volume=area of base⋅height             =∥b×c∥ ∥a∥…

Q: Use Stoke's Theorem to evaluate $. F · dR, where F (x, y, z) = (z²,xy, xz) and C is the triangle in…

A:

Q: Let x = x(4, v) be a regular parametrized surface. A parallel surface to x is a parametrized surface…

A: Let x=x(u,v) be a regular parametrized surface, A parallel surface to x is a parametrized surface…

Q: Identify the theorem: If S is the boundary surface of a solid E in 3-dimensional space, then SS, F.…

A:

Q: Let F(r, y,z) = where r2 + y? < 4, oriented by the upward normal. (r,-z,2y(r2-y²)). Find ff, F- dS…

A: Given Fx,y,z=x,-z,2yx2-y2and S is the saddle surface z=x2-y2, where x2+y2≤4, oriented by the upward…

Q: Sketch the space curve r(t) = −ti + 4tj + 3tk and find its length over the given interval [0, 1] .

A: Given space curve is rt=-ti+4tj+3tk. Sketch the above space curve as shown below.

Q: F. dS, where F(x, y, z) = z²xi + 2) + (x2z + y2)k and S is the top half of the sphere x² + y² + z? =…

A:

Q: Consider the following maps from the closed unit disk D = {(x, y) E R? : x? + y < 1} in R? to…

A:

Q: Let (x, y) and (z, w) be two points in R? state which of the following does not define metric on R2…

A:

Q: Exercise. a) For every ,x, y E R, let p(x, y) =| x? – y? |. Is p a metric on R? b) For every r, y E…

A:

Q: the following velocity relations satisfy the requirements for irrotational flow. u = xt? + 2y , v =…

A:

Q: Find the projection in the ry plane of the curve of intersection of the surfaces 222 = r² + y? and r…

A: Find your answer below

Q: Use Divergence Theorem to evaluate [0²z?i+z?x²j+x*y²k).ds, where S is the upper part of the sphere…

A: To evaluate the below integral using the divergence theorem. ∫∫Sy2z2i+z2x2j+x2y2kds Here S is the…

Q: Find the curvaturek for the space curve r(t) = (cos(t²))i + (sin(t²))j. %3D Lütfen birini seçin: a.1…

A: we have,r(t)=cost2i+sint2jhere,x=cost2andy=sint2we know…

Q: Evaluate the line integral, where C is the given space curve.     Cxeyz ds, C is the line segment…

A:

Q: The flux of F(x, y, z) = 2³î – r³}+ y®k, where o is the sphere | x2 + y? + z2 = a² with outward…

A:

Q: Let C be the portion of the circle x² + y² = 4 from (2,0) to (0,2) traced counterclockwise. (a) (b)…

A: As per the question we are given a curve C which is a quarter circle defined as : x2 + y2 = 4  ,…

Q: Use the Divergence Theorem to evaluate F. dS , where F(r, y, z) = (2x(y - z), x? – y, z) and S is…

A:

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

A:

Q: Surface o is a plane a +y + z = 1 that lies in the first octant. Evaluate

A: Given, surface σ is a plane x+y+z=1 that lies in the first octant. To find              ∬σ(2x+z)dS=…

Q: Let F(x, y, z) = 2z? xi + (y + tan(z))j + (2x²z – 2y²)k. Use the Divergence Theorem to evaluate F.…

A:

Q: Find the curvature of the space curve. r(t) = - 3i + (3+2t)j + (t +8)k %3D

A: The given curve    r(t)=-3i+3+2tj+t2+8k

Q: Evaluate F.dr, where F (x, y, z) = - yi + xj + z*k and C is the curve of intersection of the plane y…

A:

Q: Use this theorem to find the curvature. r(t) = 9t i + 2 sin(t) j + 2 cos(t) k

A:

Q: Lek F =lx?+ y?) j. Let S be the closed Surface surounding the rectangula box with vertices: %3D…

A:

Q: Sketch the space curve r(t) = ⟨4t, −cos t, sin t⟩ and find its length over the given interval [0,…

A: Consider the given space curve, rt=4t,-cost,sint use the graphing utility to graph the curve,

Q: Consider the following space curve r (t) = t²i + 2tj – v2 43/2 k %3D 3

A: The given space curve isrt=t2i^+2tj^-423t32k^.To find:The time period when rt intersects xt+zt=0 and…

Q: Verify that (S, d) is a metric space. 10.2 Prove that d(x, y) < d'(x, y)s/2 d(x, y) for all points'…

A: To prove dx,y≤d'x,y≤2dx,y for all points x,y∈ℝ×ℝ where d is the usual metric and d' is the…

Q: Evaluate J CurlF nds F(x,y,z) = yi – xj - zk, and S is part of the sphere z =V9-x2 - v? and n is…

A:

Q: Determine whether the statement is true or false. If D is the disk given by x2 + y2 < 4, then I V4-…

A: Given the given relation is

Q: Find the curvature K for the space curve r(t) = (/3sint)i + (sint)j+ (2cost)k.

A: Let, γ(t) be a space curve. The curvature of the space curve is given by, κ=γ'(t)×γ"(t)γ'(t)3.

Q: Using the Divergence Theorem, evaluate F.NdS, where F(x,y, z) = (z³i – x³j+y°k) and S is the sphere…

A: Divergence theorem relates surface integrals and volume integrals. By using the Gauss divergence…

Q: d(r, y) 1+d(x,y) 5. Let (X, d) be a metric space. Define f : X x X → R by f(r, y) = Show that f is a…

A: (χ , d)  be a metric space , define f :χ×χ→ℝ by f(x , y)=d(x , y)1+d(x , y) Positive definite…

Q: Verify the divergence theorem for A = R²ag + Rsin0ag and S is the surface of a quarter of a sphere…

A: The geometry is as follows: