Let SC R3 be a regular surface. Let A : R3 → R³ be an invertible linear transformation. Prove that AS = {Ap E R° | pE S}, is a regular surface. HINT: This is best done directly from the definition.

Q: In AMNP, T is the centroid. If MR=30 find MT. N R T M S

A: We are given that T is the centroid and MR=30. We know that the centroid divides each median in a…

Q: In AABC, X is the centroid. If CW = 15 , find XW. %3D

A: Given,           In △ABC, X is the centroid and CW¯=15

Q: Consider AABC with its centroid at point G and median at BH. What is the length of GH if BG = 3x + 4…

A: Given Data: The length of BG is: BG=3x+4 The length of GH is: GH=x+10 The ∆ABC with its centroid at…

Q: 6. For any closed surface S, prove that || CurlF.N dS=0.

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Q: If F: R3 → R3 is a diffeomorphism and M is a surface in R3, prove that the image F(M) is also a…

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

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Q: Sketch the region onto which the sector r< 2, t/4<0Sa/2 is mapped by the transformation w= = iz?.

A: 5) In this question, we have sketch the region in the w-plane by using the region in the z-plane.

Q: Let x: R² → R' be the mapping x(u, v) = (u + v, u – v, uv). w that x is a proper patch and that the…

A: Show that x is a proper patch:   A mapping defined from x : R3 to R2 is defined a patch if x maps…

Q: Use the given transformation to evaluate the integral. (8x + 8y) da, where R is the parallelogram…

A: Given that vertices of parallelogram is −3,9,3,−9,4,−8  and −2,10 And transformation is x=14u+v  and…

Q: 10) a) What is a rigid motion (isometry)? 11) Are the two figures congruent? Explain why or why not…

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Q: Let M be an elliptic paraboloid z = y² + 2?. (A) "Compute the Gauss map and the shape operator. (B)…

A: Answer for sub question a: Given x=y2+z2 A parametrization is given by: Xu,v=u2+v2,u,v Then,…

Q: Use the given transformation to evaluate the integral. || 2 Y dA, where R is the parallelogram…

A: Given, ∫∫Rx-4y6x-ydA Where R is the parallelogram enclosed by the lines x-4y=0,x-4y=2 and…

Q: Use the given transformation to evaluate the integral. (12x + 12y) dA, where R is the parallelogram…

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Q: Given, (2z – z) dz along the directed line segment from - i to 1. What is a parametric form of z(t)?…

A: We have to form a line segment from -i to 1 Let 0≤t≤1

Q: Let E The orthogonal projection of y on H = span{ u 1, ú 2} is given by:

A: Given

Q: Find the image of the trapezoid in the uv-plane bounded by the lines v = –u, v = 2 – u, v = u, and u…

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

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Q: Let C be the positively oriented square with vertices 0, 1, 1 + i, i. Evaluate using the definition…

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Q: EXERCISE 4.3. Let S be a connected regular surface such that the image of the Gauss map is a great…

A: The answer is given below thank you for the question upvote is really appreciable

Q: Let G(u, v) - (8u + v, 5u + 5v) be a map from the uv-plane to the xy-plane. Calculate Jac(G).

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Q: The transfomation takes points from the xy plane to v= 3x Jx=? that takes points from the uv plar V=…

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Q: What is the projection of r(t) = ti + t4j + etk onto the xz-plane?

A: Given: r(t)= ti+t4j+etk

Q: Let W be the region that is cut out from r² + y² < 4 by the paraboloid z = r² + y² and the plane z =…

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Q: If S is a surface parameterized by r(u, v) = (3u + 2v)î + (v – u)ĵ+ (8 – v²)k con u € [0,1], v € [2,…

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Q: Use the liner transformation T: (u, v) → (x, y) given by 1 1 T(u, v) = (-u + v)i + ÷(u+ 3v)j 4 4 to…

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Q: Let M be an elliptic paraboloid a = y? + z². Compute the Gauss map and the shape operator. Find k…

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Q: Use the flux form of Green's Theorem to evaluate |(6xy + 8y°) dA, where R is the triangle with…

A: Greens' Theorem : Let D be the region bounded by the curve C. Let F(x,y)=P,Q be the vector field…

Q: Let M⊂R ^ 3 be a surface and f be a diff.bilir function on this surface. Show that if Q is this…

A: Given: Let M⊂R3be a surface and let f be a differentiable function on this surface. If Q is a 1-1…

Q: Consider D to be the region below the paraboloid z = 3+x² + y² , inside the cylinder x + y² =1 and…

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Q: Use the Green's theorem to calulate So (3x²y – xy? – 3.x²y²) dx + (x³ – 2.x³y+ x²y) dy, where C is…

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Q: Use the given transformation to evaluate the integral. (3x + 12y) dA, where R is the parallelogram…

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Q: Use the given transformation to evaluate the integral. 15) x = 6u, y = 9v, z = 2w; x2 y2 z2 dx dy…

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Q: Calculate ff A.n ds If A = yi + 2xj – zk and S is a 2x + y = 6 plane surface in the first octane cut…

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Q: Given F=. By using Gauss's theorem, find f F ds for a paraboloid x = 4-y² - 2² with x ≥ 0 as shown…

A: We are given  F=-3x+1+z3,y+1+x3,6z+xy So, the divergence of F is div…

Q: Use the given transformation to evaluate the integral. (12x + 12y) dA, where R is the parallelogram…

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Q: Locate and classify the stationary points on the surface z=xy2-2xy-2x2-3x, remembering that  x, y ,…

A: Solving for stationary points of given function using partial derivatives

Q: Let E be the region in the plane between the two ellipses 4x² +9y? = 1 and 4x2 + 9y2 = 4. Evaluate…

A: Given that two ellipse 4x2+9y2=1 and 4x2+9y2=4 Ans given E be the region in the plane between these…

Q: Use the Jacobian of a coordinate transformation to find the area of the parallelogram bounded by the…

A: We use the following transformation to find the area of the parallelogram bounded by the lines y =…

Q: Let P be the tangent plane to the surface z=2x2+y2 at the point Q=(1,1,3). Which of the following…

A: tangent planes to a surface are planes that just touch the surface at the point and are “parallel”…

Q: Let h, k, m, neR such that n > m >0. A general equation corresponding to the above surface is: A) (z…

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Q: Consider the surface S given by: S :r(u, v) = (uv², u + v, v³), u E [0, 1], v E [0, 2]

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Q: Let G(u, v) = (5u + v, 3u + 3v) be a map from the uv-plane to the xy-plane. Calculate Jac(G).

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Q: Give an example of a parametrisation of the unit circle, C = {(x,y) E R² |x²+y² = 1}, that generates…

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Q: Let F = -2yi + 2xj. Use the flux form of Green's Theorem to compute F·n ds where C is the positively…

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Q: Let B be the region in the first quadrant of the ay-plane bounded by the lines r+ y = 1, x + y = 2,…

A: Consider the given information.

Q: Let H be the plane 2 X- 2 y- z=0 in R³, that is, H={(x, y, z)eR³ | 2x- zy- z=0} and let F be the…

A: Given that H=x,y,z∈R3|2x-2y-z=0 And, F(x,y,z)=2x-2y-2z2y+zy-z The aim is to find F(H).

Q: Let T : R³ → R³ be a surjective linear map. What can you say about the dimension of ker(T) ?

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Q: b) Use the transformation u = 2x – y, v = x + 3y to find JfR dA, x+3y where R is the rectangular…

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Q: If S is an arbitrary closed surface covering a V and A = axi + byj + czk, then www prove that f. A.n…

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Q: Evaluate z dv, where E is enclosed by the paraboloid z = x² + y2 and the plane z = 16.

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