Q: If (a,ß.r) is a point at which the surface x2+y2-z-2x+42=0 has a horizontal tangent plane, then Iyl=
A: Horizontal tangent plane
Q: Which of the following planes is tangent to the sphere x² + y + z' = 17 at the point (2, -2, 3)? %3D…
A:
Q: 18. Evaluate J (V x F) · N dS where F = yi + (x- 2xz) j- xy k and S is the surface of the sphere x…
A: Stokes theorem is used
Q: An equation of the tangent plane to the surface z = 4xy² + 3x²y² at the point (1, 1, 7) is: Select…
A: z= 4xy2+3x2y2F(x,y,z) =…
Q: 18. Let f(x, y, z) = 2x + 4y – 4z. Find the minimum and maximum values on the sphere x2 + y² + z² =…
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Q: 1. Verify Stokes' Theorem for F = (2x - y)i - yz²j - y² zk, where S is the upper half surface of the…
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Q: 28. Show that the circulation of F(x, y, z) = (x², y², z(x² + y²)) around any curve C on the surface…
A: Given figure Stoke's Theorem Here take single integral over closed path C and double integral over…
Q: the surface z = &x + 4y th=
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Q: Sketch the space curve r(t) = ⟨2 sin t, 5t, 2 cos t⟩ and find its length over the given interval [0,…
A: The given equation of space curve is r(t) = ⟨2 sin t, 5t, 2 cos t⟩.
Q: Let H be the segment of the hyperbola x-y°= | Connecting (1,0) to (2, J3). Evaluate the line,…
A: Given H be the segment of hyperbola x2-y2=1 connecting the points (1,0) to (2,3) we have to…
Q: 2. The task is to determine the flux of B across G, where G is the positively-oriented portion of…
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Q: find the absolute minimum and maximum of f(x,y)=2x^3+y^4 over the disk x2+y2 ≤ 1
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Q: If (a,B.y) is a point at which the surface x2+y2-z²-2x+12=0 has a horizontal tangent plane, then y|…
A: As per guidelines, we will solve first question only. Let α,β,γ be the point on the surface…
Q: Let V be a connected neighborhood of a point p of a surface S, and assume that the parallel…
A: Let the two points are p and q p, q∈S And the two curves are: α:I→S β:I→Sα0=p=β0αt1=q=βt2 And the…
Q: Prove that the circle {(x, y), x² + y² = 1} is hot homeomorphic to the interval [-1, 1].
A: We will prove this result by contradiction: Suppose there exists a homeomorphism between -1,1 and S…
Q: An equation of the tangent plane to the surface z = In(x - 2y) at (3, 1,0) is given by z = x + 2y- 1…
A: This question is related to three dimensions geometry.
Q: Find the absolute maximum and absolute minimum of f(x, y) = yx^2−y^2+4 on the unit disk (x, y) :…
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Q: 3. Consider Hhe following two surfaces. 8x-54-82 = -13 a) Show that the surdaces intersect at the…
A: Given surfaces are z=2 xy2 and 8x2-5y2-8z=-13 (a) Let fx,y,z=z-2 xy2=0 Let gx,y,z=8x2-5y2-8z+13=0…
Q: 4. Evaluate fo F• dr by the Stoke's theorem, where F = -y? î+x j+ z² k, and C' is the curve of…
A: Given :
Q: Suppose f(x, y) satisfies the basic existence and uniqueness theorem in some rectangular region Rof…
A: Consider the given: Let the function f(x,y) satisfies the basic existence and uniqueness theorem in…
Q: 59. Find all points on the portion of the plane x+y+z35 in the first octant at which f(x, y, z) = xy…
A: We have to find all points on the portion of the plane x+y+z=5 in the first octant at which f(x, y,…
Q: (a) Between 1 and the disk x²+ (1+x² +y*) (b) Between % = 1 and the entire xy-plane. (1+x* + y?)*
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Q: A surface x(y, z) is defined implicitly xy + yz +x'z' = 3-xZ Calculate ду
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Q: Find the absolute maximum and minimum values of the function f(x,y)=e** (x² +2y³) on the set D,…
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Q: (a) Two surfaces are called orthogonal at a point of intersection if their normal lines are…
A: To show that two vectors are orthogonal, show that the dot product is zero.
Q: Verify Stoke 's Theorem for F = (x² +y – 4) î + 3 xyj + (2 xz + z²) k over the surface of hemisphere…
A:
Q: Evaluate V x F · N dS using Stoke's theorem, where F = (2x – y)I- yz }-ýzk and S is the upper half…
A:
Q: Find an equation of the plane tangent to the surface z = x2 - y at the point (2, 1, 3). O A. 4x - 2y…
A: Find an equation of the plane tangent to the surface z= x2 - y2 at the point (2,1,3). Here i…
Q: 9. Show that the sum of the T-, y-, and z- intercepts of any tangent plane to the surface VI + Vỹ +…
A: Given, Gradient of the surface at any point (x0,y0,z0) is given by,
Q: 1) Consider the solid Q in the first octant, generat surfaces: z = (y – 3)² + 1, 2r + 4z = 16, 1= 1,…
A: Volume of solid: Let z=f(x, y) and z=g(x, y) be the functions of x and y of a solid, and let x=a to…
Q: Integrate F =-(y sin z)i + (x sin z)j + (xy cos z)k around the circle cut from the sphere x2 + y2 +…
A:
Q: The tangent plane to the graph of the surface z=In(x2+y2) at the point (1,0,0) contains point…
A: Given function is, z=lnx2+y2
Q: 14. Under the paraboloid z=x² +y2 and above the disk x +y² <9 for (x,y) in the first quadrant.
A: we have to find the volume
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: 3. (a) Show that the two surfaces S₁ z = xy and S₂ : 2 = x² - y² intersect perpendicularly at the…
A: Introduction: The gradient of a surface is a vector quantity that is directed in the normal…
Q: Sketch the solid described by the given inequalities. 0srs 8, -T/2 se S T/2, 0szs 1. 1.0 1.0 z05…
A:
Q: Find the surfaces on which the function u = 2 (x2 + y²)/z is constar
A: Since u is a constant so, x2+y2z=4a , where a is an arbitrary constant.
Q: find the flux of f=zk across the portion of the sphere x2+y2+z2=a2 in the first octant in the…
A: Given the field, F=zk The surface S is the portion of the sphere x2+y2+z2=a2 in the first octant in…
Q: Find the absolute maximum and absolute minimum of f(x, y) = yx² – y² +4 on the unit disk {(x, y) :…
A:
Q: Plot the implicitly given surface (x − 1)² /4 + (y − 1)² +z² = 1 (a) using the sphere function. (b)…
A: As per the question we have to plot the following surface : (x-1)2/4 + (y-1)2 + z2 = 1 With the help…
Q: Find an equation for the tangent plane to the surface z = 3y^2 − 2x^2 + x at P(2, −1, −3). Express…
A: Let's find equation of tangent plane.
Q: Show that the tangent plane to the surface x 2 a 2 + y 2 b 2 = cz at the point P(x0, y0, z0) is…
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Q: 5. Prove that the surface E, = {(x, y, z)|a² + y² = 4} is locally isometric to the surface E2 = {(x,…
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Q: verify Stokes § = (a%-y)i - yz'i over the uppea hazf SOD Face Of the sphere =1 bounded by the
A: Given that, F→=(2x-y)i→-yz2j→-y2zk→ To verify Stoke's theorem: ∫CF→·dr→=∬Scurl F→·n^dS Here C is…
Q: Which of the following is the equation of the tangent plane to the surface z=x2 - x+3y² +y at the…
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Q: Let y ==a and z =. Find the intersection of these х x2 surfaces.
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Q: (x - y) dv, where E is enclosed by the surfaces z = x² - 1, z = 1 - x², y = 0, and y = 4
A: Consider the given function fx,y,z=x-y. The region E is enclosed by the surfaces…
Q: *6. Let f: R' →R be given by =+4x'z-6xyz+4y-3x'y. Is M=f-((0}) a smooth surface (2-dimensional…
A: Given, fxyz=z2+4x3z-6xyz+4y3-3x2y2 Now, writing the given function in matrix form.…
Q: The tangent plane to the graph of the surface z=In(x2+y²) at the point (1,0,0) contains point…
A:
6
![(3) Prove that the two Riemann surfaces D: {z e C:|z| < 1} and H:= {w€C:
I(w) > 0} are equivalent.
(4) Prove that the surfaces T2HRP2 and RP2 HRP²#RP² are topologically equiv-
alent. Cf. [Donl1, Chapter 2, Lemma 1].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fade615cf-0b36-402b-afc2-ec9428495368%2Fb2bab785-1baf-4973-ba5c-f38d49368334%2Fayl4ggk_processed.jpeg&w=3840&q=75)
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- Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?Classify the quadric surface. z = y2 − x2 16The points on the surface x2−2y2−3z2 = 33 at which the tangent plane is parallel to the 8x+4y+6z=5 plane are:a. (1,0,1) y (2,3,1)b. (-4√3, √3, -√3) and (-4√3, -√3, √3) c. (4√3, -√3, -√3) and (-4√3, √3, √3) d. (-4√3, -√3, -√3) and (-4√3, -√3, √3)
- Find a parametrisation of the curve of intersection of the surfaces: x^2 + 2y^2 + z^2 = 5 and x^2 + y^2 = 1 which lies in the first octantCompute the intersections of the curve xy = 1 and the lines x +y = 5/2, x+y = 2, x+y = 0, x=0 , x=1 in the affine space and then in the projective space by using homogeneous coordinates. Complex solutions are valid. Please show your steps for both affine space and in project space. Box your final answer.Find the extreme values of ƒ(x, y, z) = x2yz + 1 on the intersection of the plane z = 1 with the sphere x2 + y2 + z2 = 1
- Sketch the surfaces CYLINDERS z = y2 - 1Two surfaces S and S^(-) with a common point p have contact order ≥ 2 at p if there exist parametrization x(u,v) and x^(-)(u,v) in p of S and S^(-) respectively such that xu = x^(-)u, xv = x^(-)v, xuu = x^(-)uu, xuv = x^(-)uv, xvv = x^(-)vv at p. Prove the following: a. Let S and S^(-) have contact order greater than or equal to 2 at p; x:U -> S and x^(-): U -> S^(-) be arbitrary parametrizations in p of S and S^(-) respectively and f: V c R^(3) -> R be a differentiable function in a neighborhood V of p in R^(3). Then the partial derivatives of order smaller than or equal to 2 of f o x^(-): U -> R are zero in x bar^(-1)(p) iff the partial derivatives of order smaller than or equal to 2 of f o x: U -> R are zero in x^(-1) (p). b. Let S and S^(-) have contact of order smaller than or equal to 2 at p. Let z = f(x, y), z = f^(-) (x, y) be the equations in a neighborhood of p, of S and S^(-) respectively where the xy plane is the common tangent plane at p = (0, 0). Then the…Suppose that the Celsius temperature at the point (x, y, z) on the sphere x2 + y2 + z2 = 1 is T = 400xyz2. Locate the highest and lowest temperatures on the sphere.