The first fundamental form of the surface (S) is given by I(du, dv) = ud?u + 4vdudv + d²v Consider the transformation 0 = u – v, o = u. Then I(d0, dø) = d*e + (40 - 14 - 2)dedo + (1- 18 + 5ø)d*o (1- 40 + 5)d'0 + (40 - 10 - 2)dodo + d'+ The a bove answer The above a ns wer d28 + (1- 40 + 5o)dodp + (40 - 14 - 2)d+ d0 + (40 - 44 - 2)dodo + (1 - 40 + 5p )d*¢ The above answer The above a ns wer

Q: 8. Consider the surface S of equation z²-2x² + y²-x= 0 and let P (2, 3, 1) be a point at S. Suppose…

A: In this question, concept of gradient is used. Gradient A scalar field's gradient may be a vector…

Q: Apply the transformation и %3D 2х + у, v %3D 3х + 2y to the region 13x? + 16xy+5y² < 4. Describe the…

A: First of all we try to find x and y in terms of u and v. There after we transform the given region…

Q: 2. A surface S in R®(r,y, :) is defined parametrically in the form u + -1 2u (a) Find points on the…

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Q: Find the point(s) on the surface z2 = xy+1 which are closest to the point (10, 8, 0). List points as…

A: The objective is to find the points on the surface z2=xy+1 which are the closest to the point 10,8,0…

Q: (a) Evaluate S, dV, where E is the solid enclosed by the y2, z2 2+ = 1. Use the transformation x =…

A: Here we solve the given calculus problem.

Q: calc 3 13.7#1 make sure its right Evaluate the surface integral. ∫∫S yz dS S is the surface with…

A: Given a surface with parametric equations, x=u2, y=usin3w, z=ucos3w, 0≤u≤1, 0≤w≤π6. To evaluate the…

Q: The base of the closed cubelike surface shown here is the unit square in the xy-plane. The four…

A: Checking the gradient of the function: Thus, the net outward flux should be 0, The sum of flux…

Q: Find a transformation that maps R = {z = x + iy : x, y 2 1} onto the unit disk |2| < 1.

A: please see the next step for solution

Q: A 2-D surface was found in one research group of scientists and a surface equation has been derived…

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Q: 16)Consider the equation of the surface S given by In (xyz) + xyz + x3 - y3 + z3 = 2 An equation for…

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Q: 3. Find the area of the part of the surface x 2z + y° - 3, that lies above the triangle in the…

A: Given, x=2z+y2-3, i.e. z=12x-y2+3. As it is known that the surface area is given as,…

Q: Using Green's Theorem, find the outward flux of F across the closed curve C. F = In(x² +y* )i + tan…

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Q: Determine the projection of the following surface on the xz plane. (xz trace) 4x – + = 0

A: Given equation of surface is 4x-y24+z28=0

Q: 6. Use Stokes theorem to evaluate $. F•dr, where F = (-3y² , 4z, 6x) and C is the triangle in the…

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Q: Let C is a line segment from (3,0,1) to (0,3,9) with F ={(x+1, y,=). The following table provides…

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Q: Evaluate / . F. dS Is where F = and the surface S is the triangle with vertices (1, 1, 0), (0, 2,…

A: Given To evaluate ∬SF.dS  where F=&lt;x+1,-1-2y,z&gt; and the surface S is the triangle with the…

Q: Sketch the region in the plane consisting of points whose polar coordinates satisfy the given…

A: The region 3&lt;r&lt;4, 5π3≤θ≤7π3. We have to sketch this region.

Q: 9. Evaluate F - dS, where F(x, y, z) = xy²i+x²yj+y² k, and S is the positively oriented surface…

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Q: A = xy'a, + y'a, + y°za, and S is the surface of the cuboid defined by 0 < x < 1, 0 < y < 1,0 < z <…

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Q: The parametric representation of the surface y2 - 4y + z2 = 0,0 < x < 3 is a) 7(u, v) = (v, 2 + 2…

A: solution:y2-4y+z2=0 , 0≤x≤3considery2-4y+z2=0∴y2-4y+4+z2=4∴y-22+z2=22

Q: Let D be the solid bounded by the ellipsoid x2/a2+y2/b2+z2/c2=1 wher a, b,c>0. Let T be the…

A: Given a solid D bounded by an ellipsoid x2a2+y2b2+z2c2=1, where a, b, c&gt;0. Let T be a…

Q: Consider the transformation x =r cos 0, y = r sin 0, z = z from cylinderical to rectangular…

A:

Q: Sketch the surfaces HYPERBOLOIDS (y2/4) - (x2/4) - z2 = 1

A: we have to sketch the surface of the hyperboloids  y24-x24-z2=1

Q: 24. Orthogonal projection onto the plane x + 2y + 3z = 0 in R transformation T from R³ to R' given…

A:

Q: 1. Let a, b, c ER and let Sa,b,c = {(x, y, z): ax2 + by2 + cz²=0, x² + y² +2²=1}- (1a) For which…

A: Given: Sabc=(x,y,z): ax2+by2+cz2=0, x2+y2+z2=1 To find: the values a, b, c for which Sabc a regular…

Q: (a) Write the expression for the Jacobian of a transformation T from uvw-space to xyz-space defined…

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Q: The parametric representation of the surface x² – 6x + z² = 0,0 < y < 2 %3D is a) (u, v) = (3 cos u…

A: The general form of the parametric equation of the surface in three-dimensional space is given by…

Q: 2. A surface S in R'(z,y, =) is defined parametrically in the form u + -1 -( ) X(u, e) = 2u 2 (a)…

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Q: Evaluate the surface z =1,0<x,y <1.

A: To solve this we will use divergence theorem.

Q: Let F (tanx + y)ĩ + (x + secy)j and C be the boundaries of the ellipse 9y=9 in the counterclockwise…

A: Given - F→ = tanx + yi^ + x + secyj^ and C be the boundaries of the ellipse x2 + 9y2 = 9 in the…

Q: 3. Consider the surface z = xln y – x²y. (a) Show that Q = (1,1, –1) lies on this surface. (b) Find…

A: The given surface is z=xlny-x2y (a) Substitute (x,y,z)=(1,1,−1) in z=xlny-x2y.…

Q: 14. Consider the surface S: z = 2ev - xy, where P (-1, 0, –1) is a point in S. An equation of the…

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Q: Evaluate / . F. dS where F = and the surface S is the triangle with vertices (1, 1, 0), (0, 2, 0),…

A: First, we plot the surface S, and try and form its equation.Since S lies in a plane, it is the part…

Q: (a) Evaluate fl, dV, where E is the solid enclosed by the x2 ellipsoid q2 z2 = 1. Use the…

A: (a) Here, we have x=a·u, y=b·v and z=c·w. The Jacobian is as follows: ∂(x, y, z)∂(u, v,…

Q: A rectangle ℛ with sides a and b is divided into two parts ℛ1 and ℛ2 by an arc of a parabola that…

A: A rectangle ℛ with sides a and b is divided into two parts ℛ1 and ℛ2 by an arc of a parabola that…

Q: Consider the surface S shown in the graph, whose parameterization is given by: r(u, v) = (v², 2u²,…

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Q: (a) Find a transformation which removes the zy term in z+zy+y = 16 and (b) give a geometrie…

A: To remove the product, there needs a rotation of axes about the origin.

Q: Use the transformation x = * and y = uv (with u > 0 and v > 0) to evaluate /| xy dA R where Ris the…

A: The integral is ∬Rxy2dA. Transformation: x=uv and y=uv We obtain the integral as follows:  ∬Rfx,…

Q: Given: CB I ED; CB ED Prove: ACBF AEDF using isometric (rigid) transformations. E D B

A: Given two triangles, △CBF and △EDF . Also, CB≅ED and CB∥ED. In △CBF and △EDF , Given that, CB≅ED…

Q: 2. Find the exact area of the surface. a. z = 1+ 2x + 3y + 4y², 1 < x < 4, 0 < y < 1. b. z = 1+x +…

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Q: 13) Parametrize the boundary curve of the surface given by r(r, 0) = (r cos 0, r sin 0, (4- r2)) for…

A: The parametric equation of the surface is given as follows: rr, θ=r cos θ, r sin θ, 4-r2, where…

Q: (a) Use the transformation r = 7u + v, y = u+ 7v to rewrite |/ (x – y)dA, where R is the triangular…

A:

Q: The equation of the tangent plane to the surface f (x,y,z) = xecosz - z - 1= 0 at the point…

A: The answer is as follows :

Q: Which of the following is a defining equation for the tangential plane to the surface defined by…

A: To find  An equation for the tangential plane to the surface defined at the point (1, 1,1)

Q: The parametric representation of the surface y2 - 4y + z2 = 0,0 <x< 3 %3D is a) F(u, v) = (v, 2 + 2…

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Q: 33. Figure 14.13 shows the saddle-shaped surface z = S(x, y). (a) What is the sign of f,(0, 5)? (b)…

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Q: Suppose F(x, y) = eW5} – sin(x)} nd C is the counter-clockwise oriented rectangle with vertices (0,…

A:

Q: The parametric representation of the surface x2 -8x + z2 = 0,0 < y 5 5 %3D is a) (u, v) = (4 cos u ,…

A:

Q: (a) Evaluate JE DV, where E is the solid enclosed by the ellipsoid x²/a²+ y²/b ² + z²/c² = 1. Use…

A: Given that the solid enclosed by the ellipsoid x2a2+y2b2+z2c2=1

Q: Find a Möbius transformation which maps the region outside the unit circle onto the left- half…

A: We have to find a Mo¨bius transformation which maps the region outside the unit circle onto the…