8. Consider the surface S of equation z²-2x² + y²-x= 0 and let P (2, 3, 1) be a point at S. Suppose z is an implicit function of x and y in a neighborhood of point P. Consider the following statements: III. The vector Vz (P) is unitary. Is this true or false?

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: c) Show that the vector field W(x, y, z) = (2yz+ xz – x², –2xz – yz + y?, y² – x² + z²) - - is a…

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Q: 4. Find the positively oriented flux over the surface created by the paraboloid z = 16 – x² – y2…

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Q: 9. Show that the following vector fields are conservative and calculate ( É ·ds for the given curve…

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Q: 1. Let S C R be the surface defined for all 0 < x < 1, -1 < y < 1 by z = r² + y. Compute ffsa ds

A: It can be solved by partial differentiation and integration

Q: (a) Prove that M: (x² + y^y° + 3z² = 1 is a surface. (b) For which values of e is M: z(z – 2) + xy =…

A: Consider the provided question, (a)

Q: If C is the curve given by r (t) = (1+ 5 sin t) i+ (1+3 sin? t) j+ (1+3 sin t) k, 0 <t< and F is the…

A:

Q: 25. Evaluate F.N dS where F= yl + 2xj –- zk and S is the surface of the plane 2x +ye 6 in the first…

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Q: Given a vector field F = 5(x² + y²)i+ 2xyj - 3zk, evaluate SSS, V· FdV for which the solid D is…

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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: 15) Consider the vector field F (x, y, 2) = (In y, x, z) and the oriented curve C:r (t) = (t, e', 2…

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 2 + x² + y°, the cylinder z² + y² = 2, and…

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Q: 1 : : Evaluate F dS whereF = (r",y",) and S is the solid bounded by the ry plane and the elliptic…

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Q: Exercise 10.2 Evaluate the line integral fc(² – 2y²) ds, where C is the line segment from (0,0) to…

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Q: 8) The vector field Vp[x, y] = F =- has a potential function p[x, y] = a) (x² + y²)³2 b) d) Vx? +y?…

A:

Q: Find the flux of F = yi + 2zxj – z²k through the parabolic cylinder y = x2,0 < x < 1,0 < z< 4, in…

A:

Q: 5. Find the outward flux of the vector field F(x, y,z) = x²i + y°j+z²k through the first octant…

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Q: (b) Given a vector field F = 5(x² + y²)i+ 2xyj – 3zk, evaluate JI, V· FdV for which the solid D is…

A: Here we use polar form of circle .

Q: Example 3 a) Evaluate the surface integral SaS 2x²y dS over the surface y² + z² = 1 between %3D x =…

A: The given problem is related with surface integral. We have to evaluate the surface integral…

Q: Carry out the affine transformation | u = (x + y)/V2 lv= (y - x)/V2 in the improper double integral…

A:

Q: 5) Find the work done by the force field F(r, y, z) = ri-zj+y k in moving an object along the…

A: Work = Force X Displacement

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: (c) Determine the flux of the vector field F = x³î – 2yĵ + 7z²k across a cone z = Jx2 + y2 + 3 that…

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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,…

A:

Q: (c) Determine the flux of the vector field F = x³î – 2yĵ + 7z²k across a cone z = J 8 by using…

A:

Q: 4. Show, in detail, that any vector field of the form (*,J, 2) = J (*) +8(y)/+h(z)k is irrotational,…

A: To show that the vector field is irrotational.

Q: 4. Determine whether or not the vector field F(x, Y, z) = (-xye®, -xe + e° + 2y, 2z) is…

A:

Q: 3.4.8 Show that the sphere x² + y² + z² = 1, z ≥ 0. JJ,z²dS=, where S is the surface of

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Q: Given that the vector field F(x, y) = (2x-3y, 2y-3x) is conservative in the xy-plane. If (x, y) is a…

A: We have to find the value

Q: Suppose S is the surface where x² + y² + 3z² = 1. Let F be the vector field F = (xy, yz, z²).…

A: We need to evaluate integral.

Q: A path C is defined by the parametric equations x(t)-t, y(t) = 2t, 0≤t≤ 1, and a vector field is…

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Q: I have trouble getting to the answer. Just wondering if I can get it step by step.

A:

Q: 3. Let H = 2xi – 3xy j+xz² k. Calculate ſ, Ĥ •d where S is the surface of the cube with corners at…

A: Given- Let H→ = 2xi→ - 3xyj→ + xz2k→ To find- Calculate ∫SH→·dA→ where S is the surface of the cube…

Q: A path C is defined by the parametric equations z(t)=t, y(t) = 2t, 0<t<1, and a vector field is…

A: It is given that a path C is defined by the parametric equations x(t)=t, y(t)=2t, 0≤t≤1 and a vector…

Q: In R, let S be the piece of the plane 3x + 3y + 3z = 3 that lies in the first octant, oriented…

A: Given S is the piece of the plane 3x+3y+3z=3 that lies in the first octant oriented upward. The…

Q: Show that the vector field F = 2.xyi +(x² – 2 y)j is conservative and evaluate the line integral…

A: Given the vector field

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: Suppose that the vector field F(2, y, z) = (4y – 2z)î + (3y – 4z)j + (5æ + y)k and let the surface S…

A: Given a vector field      F→x,y,z=4y-2zi→+3y-4zj→+5x+yk→. The surface S be the part of the cone…

Q: 7. Consider the surface S defined by the equation x³ +y2 – 3x – 2y – 4z = -2. Find all points P =…

A:

Q: Example 6: Evaluate ez dz dy dx E Where E is enclosed by the paraboloid Z = 3+ x? + y² , the…

A: The given problem is to evaluate the given triple integral of given function ez for which region E…

Q: 4. Calculate the flux of F = over the square shown below, oriented counterclockwise. SS (Px +Q, R…

A: We have to check given work on this problem write or wrong   But this work is wrong Because flux or…

Q: 1. Calculate the work done when a force F = 3xyî – y?j moves a particle in the xy-plane from (0,0)…

A: Take dot product along limit of integration and integrate

Q: Find the Duf of f(x, y,z) = z In(x+ y) + e3y on the vector u = 3i -j-4k at Po(1,0,1)

A:

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: Ex.5 Given the Jordan curve y that is the counterclockwise oriented boundary of the set T= {z=x+iy :…

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Q: Find a conformal map f (z2) which takes the disk {z : 2 < 3} onto the left half plane {w: Re(w) }and…

A: Given clear step by step explanation below

Q: Given vector field F(x, y) = (4x³y? – 2xy³,2x*y – 3x?y² +6y³) and curve C defined by C: 7(t) = (t+…

A: Here we will find out the required value.

Q: Justify the following statement: The flux of F = (x3, y3, z3) through every closed surface is…

A: The flux through a closed surface is given by Gauss's divergence theorem, Flux, ∫∫∫∇→·F→dV=∯F→·n^ds…