Exercise 3. Prove that M = {(x, y, z) | (x² + y³)² + z² = 1} is a surface. (Hint: this surface is the level set of a function g : R3 → R).

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: c) The surface S is the portion of the plane x+y+z=1 in the first quadrant of the Oxyz Cartesian…

A: Given: The surface and curve is shown below   The region s is the triangle and the curve C is the…

Q: 4. Find the maximum and minimum values of the function f(x, y, z)-x+y-z over the sphere x² + y² +…

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Q: Find the minimum and maximum points of on the sphere and the value of f at those points. f(x, y, z)…

A: Lagrange multipliers are used to find the maximum and minimum points of the function. Minima means…

Q: Let (S) be the part of the sphere (x – 3)² + y² + (z – 1)² = 4 tha Then (S) is the graph of the…

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Q: B- Consider the following subsets of real plane R² Determine with graph: 1- Arcwise connected A =…

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Q: Explain briefly how Vø relates to surfaces of constant o(x, y, x). Hence, use Vø to find points in…

A: For the surface of constants ϕx,y,z=k, the gradient vector ∇ϕx0,y0,z0 is orthogonal to the level…

Q: Evaluate ff, (F n) dS in the following cases: (a) F = (2y+z, x+z, y − x) and S is the portion of the…

A: (a)

Q: Suppose that ŋi and 72 are two-forms on R such that Ni = 72 + dw, where w is the one-form w = (x +y)…

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Q: Match each function with the description of its f(x, y, z) = 0 level surface. e f(x, y, z) = -2² -…

A: f(x,y,z)=-x2-y2-2z2=0 This is satisfied only by the points (0,0,0). Therefore, it corresponds to (d)…

Q: 5. Consider the surface S of equation 1-2 - y² An equation for the line tangent to the surface S at…

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Q: Find the maximum and the minimum values of f(x,y,z) = 5x -8y + 8z on the sphere x² + y² + z² = -

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Q: 8. Provide a rough sketch of the surface in R³ : (a) defined by 3x² − 12x − y + 2x² + 4z + 9 = 0.…

A: We have to identify the surface and provide a sketch

Q: f(-2)=3 f'(-2) = 4 g (4) = 5 g (4) = 2 The point (-2,4) lies on the curve in the xy-plane given by…

A: f(-2)=3f'(-2)=4g(4)=5g'(4)=2

Q: F = (x² – y²,x, 2yz); C is the boundary of the plane z = 6 – 2x – y in the first octant.

A: Given, F=x2-y2,x,2yz Find the line integral ∮CF·dr in the following. Here C is the boundary of the…

Q: (3) Consider the quadric surface SC R, having equation a2 + y? – 22 = 9. Find all lines through P =…

A: The quadratic equation is given by x2+y2-z2=9.  General equation of line is: x,y,z=a,b,c+tl,m,n…

Q: Is the set {(x, y, z) = R³; z = 0 and x² + y² ≤ 1} a regular surface? Is the set {(x, y, z) = R³; z…

A: Solution:-  (a) This is not a regular surface because a neighborhood of a boundary point like (1, 0,…

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: Show that set : A = {(x,y) = R² : sin(x) ≤ y ≤ 2e-2²] is closed on Euclidean plane R².

A: The given set is: A=x,y∈ℝ2: sinx≤y≤2e-x2. We need to shoe that the set A is closed in Euclidean…

Q: a) Evaluate ff, (2x – z) dS, where S is part of the plane 3x + 2y +z = 1 that lies above the…

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Q: Is a subset of the plane {(x1, x2); x1 > 0, x1x2 ≥ 1} convex? Justify.

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Q: Let R = (c, 1,-1) where c denotes a real number. Let ä denote the plane x + y – z = 2. Find the…

A: Given Information -  The distance from the point R = (c,1,-1) to the plane x+y-z = 2 is 2. We have…

Q: Sketch the level surface of the function for w = 3. w = f(x, y, z) = x² + y²+z² + 2y

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Q: 2. Evaluate ([.(Vx F). ds, where S is the surface of the paraboloid x2 + y2 + z = 4 above the…

A: We have given F→=(x2+y-4)i+3xyj+(2xz+z2)ksurface is given x2+y2+z=4 and above the xy planediagram of…

Q: Prove that the z-axis is a nodal line with unodes a -2) and (0, 0,2/3) for the surface 3 2xy +…

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Q: Find the envelope of the plane y Z + u X a + u + + b+uc = 1.

A: Given Data: Let us consider the given data, xa+u+yb+u+zc+u=1 To Find: The envelope of the plane.

Q: Let S be the surface in R³ defined by the equation x4. y5 - 5z = 8. Find a real-valued function g…

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Q: Let C be an ellipse in a horizonal plane lying above the xy-plane. Which type of quadric surface is…

A: Given, Let C be an ellipse in a horizontal plane lying above the xy-plane.

Q: 12. a. Use vectors to show that the distance from P;(x1, Y1, Z1) to the plane Ax + By + Cz = D is…

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Q: then A(-2u - 20) = -2 ---B 5 and Au 3 Au = 2 4

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