Assume S is a smooth surface parametrized by R(u,v) = (V / za sin y do. Evaluate S Hint: Find the domain of R to obtain the bounds for u and

Q: Consider the surface defined by the following function: x? + y? 2 = Find the points on the given…

A: The equation of the surface is given as, ∅=x2+y2-z The gradient vector at a surface z is given by,…

Q: Let L be the line segment with endpoints at (1,2,3) and (-1,1,4). A light source at is at the point…

A: We will find shadow of given points on the plane given and then find parametric equation of the line…

Q: Your friends correctly calculate the gradient vector for f(x, y)=x²+y at the point (2,-3) as…

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Q: Let S be the surface defined by the vector function R(u, v) = (u cos v, u – v, u sin v) with u ER…

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Q: suppose gradient f(a,b) =4i + 3j. find a unit vector u such that a) Du f(a,b)=0 b) Du f(a,b) is a…

A: Please see the below picture for detailed solution.

Q: Q2/ Find T, N, and k for the space curve r(t) = (3 sin t)i +(3cos t)j + 4tk

A: Question: Objective is to calculate the tangent vector T, normal vector N and the curvature κ of…

Q: Let P = ( 0 , 0 , 2 ) and let S be the unit sphere with equation x2+y2+z2=1. The collection of…

A: Let P = ( 0 , 0 , 2 ) and let S be the unit sphere with equation x2+y2+z2=1.  First, we need to find…

Q: 3. Find the divergence of U3 -e3 U1U2 U1 U2 -ēj + e2 + F 2uzu3 U3 if hi = U1U2, h3 = Uz. = U1, h2

A: Given the field , F=u12u2u3e1+u2u3e2+u3u1u2e3 and h1=u1, h2=u1u2, h3=u3 Divergence of F is given by,…

Q: Verify Stoke's theorem for the vector point function yz2 (xz² .3 + ex over the surface S of the cube…

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Q: Evaluate F.dS, where F = and S is the helicoid with vector equation r(u, v) = 0 < u < 3, 0 < v < T…

A: Given that, F¯=y,-x,z5 and ru,v=u cos v, u sin v, v where 0≤u≤3 and 0≤v≤π. Find ru×rv as follows.…

Q: Assume that u and u are continuously differentiable functions. Using Green's theorem, prove that Uz…

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Q: The surfaces intersect in a space curve C. Determine the projection of C onto the xy-plane.…

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Q: Q4. For F = xy?a, + yz²a, – x²zaz, use divergence theorem to evaluate ff, F. ds where S is the…

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Q: Find a parametrization for the line segment joining the points P(10,0,10) and Q(0,10,0). Draw…

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Q: Consider the space curve represented by the intersection of the surfaces. Represent the curve by a…

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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: The equation of the hyperbola that has a center at (1, 5), a focus at ( – 4, 5), and a vertex at (4,…

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Q: Use software to plot the function f(x, y) = x² + y² with the intersecting planes x = 0 and z = 4.…

A: Perspective 1: 3D Plot of  f(x,y)=x2+y2In this view, we see the 3D surface plot of the function…

Q: sketch the space curve represented by the intersection of the surfaces. Then represent the curve by…

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Q: A normal vector to the tangent plane for the surface z = , at the point (1, 1, 1) is: y - Select…

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Q: Sketch the curve associated with the vector function r(t) = . Hint: Show it lies on the surface z?…

A: Given the vector function is r→t=tcost,tsint,t. Skecth the curve associated with the above vector…

Q: find the principal curvatures and principal vectors of the saddle surface M:z = xy at the origin.

A: To find the principal curvatures and principal vectors of the saddle surface M: z = xy at the…

Q: 2. Consider the following vector function: r(t) = = a cos wti + a sin wtj + bwtk, where a, b and w…

A: We need to find radius of curvature.

Q: R(u, v) = (2v + cos u, 2u + sin v, uv)

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Q: Let f : R2 → R differentiable with f(a, b) = 5, ∂f/∂x (a, b) ≠ 0 and ∂f/∂y (a, b) ≠ 0. Let C be the…

A: “Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: JJs where S is the sphere x² + y2 + z2 = 1.

A: Given S is the sphere                         x2+y2+z2=1 To evaluate:             ∫∫S3x+3y+z2dS…

Q: Let S be the surface defined by the vector function R(u, v) = (u cos v, u – v, u sin v) with u eR…

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Q: Please show full solution for thumbs up!

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Q: Let S be the quadratic surface given by S = {(x, y, z) | z = 4x² - y², z ≥ 0}, oriented with the…

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Q: Find T, N, and k for the space curve r(t) = (7e'cos t)i + (7e'sin t)j + 4k.

A: Given data:  r(t)=(7et cost) i +(7et sin t) j+4k