1. a) Use the cylindrical coordinates to evaluate (3 + xy)dV, where E is the solid E in the first octant that is bounded by the paraboloid z = 1-x² -y², and the coordinate planes z = 0, x = 0 and y = 0.

Q: 16. Calculate ]] (V × F) · în dS, where F = (y, −x, e²²) and S is the part of paraboloid z = 3x² +…

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Q: 3) Let à = p(z² – 1)ū, – pzcos(4)ug + p² z²Uz and B = r? cos(4)u,+ 2rsin(0)ūg At T(-3,4,1),…

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Q: Find the equation of the tangent plane at the point T(1,1,4) of the sphere (x – 1)2 + (y – 2)² + (z…

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Q: b) Given F(r, y, z) = (r³ + cosh z) i+ (2y3 – 3r*y)j – (2² + 4y²z) k. Use Gauss's theorem to…

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Q: (5) Find the area of surface of -3. (a) The part of the paraboloid z = 1 - ² - y² that lies above…

A: The given problem is to find the surface area for the given surfaces over the given region. a) Part…

Q: b) Given F(r, y, z) = (x³ + cosh z) i+ (2y - 3ry)j – (x² + 4y²z) k. Use %3D Gauss's theorem to…

A: 1. The divergence is an operator, which takes in the vector-valued function defining this vector…

Q: i. Find b × c in terms of λ . ii. Show that a · (b × c) is independent of λ .  iii. Find the…

A: i)

Q: (x, Y, 2) : 0 < ¤ < V4 – y, 0 < y < 2, Væ² + y² <z< V4 - x² y Let T = · Evaluate //I (x² + y? + 22…

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Q: Find a vector, r(t), that represents the curve of intersection of the two surfaces. The paraboloid z…

A: We need to find the vector, r(t), theat represents the curve of intersection of the two surfaces.The…

Q: Sketch the three projections of F(t) = (3 sint, ,3 cost) onto the ry-, yz-, zz-planes. Parametrize…

A: Question (1) r(t)→ = &lt;3sint, t2, 3cost&gt; The projection of a vector on a plane is its…

Q: 12. Let C represent the curve of intersection between the cylinder x² + y² = 4 and the plane z = 10…

A: The plane z=10-x will cut the cylinder x2+y2=4 to give an ellipse.

Q: The intersection of the ellipsoid x? + 4y? +z? = 36 with the plane z = V3x gives an ellipse C. A)…

A: Given the ellipsoid x2+4y2+z2=36---1 and the plane z=3x---2. The intersection of these surfaces…

Q: Consider the ellipsoid 4x2 + 36y2 + 225 z2 = 225 (a) Parametrize the surface S = r(u, v) by x = a…

A: We will answer the first two questions as we don't answer multiple questions at a time. Please…

Q: Consider r(t)=⟨sin(t),cos(t),2t⟩ the position vector of a particle at time t∈[0,3], where the…

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Q: Let G be the solid in the first octant bounded by the sphere 2x? + 2y? + 2z? = 32 and the coordinate…

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Q: a) Evaluate [-ri-yj+2: k) - n ds, (-a i- where a is the portion of the paraboloid z = 3r + 3y…

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Q: Lo the ry-plane at the point I = I, + m(x² + y) where I, is the moment of inertia about (0,0). (This…

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Q: Consider the spherical surface E of center at C= (0,0.2) and radius 3. There are two planes parallel…

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Q: Calculate the positive value of m so that the x+y+2z=m plane is tangent to the spherical surface…

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Q: Use cylindrical coordinates to evaluate SG √x²++y² dV, where G is the solid above z = 9 - x² - y²…

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Q: Let G be the solid in the first octant bounded by the sphere 2x? + 2y? + 2z? and the coordinate…

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Q: Let H = 38 + 29 – 42 a) Express H in cylindrical coordinates. b) Evaluate H at point P(2,60°, –-1).

A: Here we are given :H=3i^+2j^-4k^ Here x=3 , y= 2, z=-4 Now Converting into cylindrical coordinates,…

Q: In cylindrical coordinates, a particle has position vector F = rê, + ze = r cos(0)i +rsin(0)j + zk.…

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Q: (d) (e) x'dy, where E is the solid that lies within the cylinder x² + y² = 1 E above the plane z = 0…

A: Answer d)

Q: Use Stokes' Theorem to find m # 0 satisfying F · dr = -5, whereF(r,y, z) = (3yx² + z³, y², 4yx²) e C…

A: Given:

Q: 2. Calculate the Jacobian of the coordinate change (x, y, z) to cylindrical coordinates (r, 0, z).

A: To calculate the Jacobian of the coordinate change  to cylindrical coordinates .

Q: 4. a) Find the surface area of the part of the sphere x² + y² + z² = 4 that lies above the plane z =…

A: note : Since you have posted multiple questions, we will provide the solution only to the first…