Q#1 Evaluate ſ,A.n ds , where A = z i+ xj-3y°z k ( i, j, k are unit vectors) and S is the surface of the cylinder x² + y² = 16 included in the first octant between z = 0 to z = 5.

Q: If E is the solid that lies under the plane z = 3 which its base on the xy-plane is between the two…

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Q: S, ds, where S is the hemisphere x² + y² + z² = 1,z 2 0. The parametrization of the hemisphere is: x…

A: Given that, The parametrization of the hemisphere is:

Q: Consider the surface S: z = 4ex^2  - 4xy, where P (0, 1, 1) is a point on S. An equation of the…

A: A detailed solution is given below.

Q: If A = 2yz i - (x +3y - 2)j + (x²+z)k, evaluate |(V×A) •n dS over the surface of intersection of the…

A: Given:  A=2yzi-x+3y-2j+x2+zk, With the cylinder x2+y2=a2,x2+z2=a2 Now, calculating ∇×A.…

Q: y dV, where B is the portion of the unit ball B = {(x, y, z) : x² +y² + z² < 1} which lies below the…

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Q: 1. Evaluate the surface integral fs (r + y + z)dS, S is the parallelogram with parametric equations…

A: We are entitled to solve only one question at a time. As the question number is not mentioned, So we…

Q: ds, where S is the hemisphere x² + y² + z² = 1, z ≥ 0. The parametrization of the hemisphere is: x =…

A: Introduction: Surface integral is sometimes evaluated using parametrization. It helps to reduce the…

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: Evaluate fs F dS, where F(r, y, z) = (2², xy, z) and S is the part of the paraboloid z = 22 + y?…

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Q: Evaluate | (3z2 – 22) dz, where C is parametrized by z(t) = t+iť, 0<t< 1. -

A: Given integral is ∫C3z2−2zdz ,where C is parametrized by zt=t+it2, 0≤t≤1. We have to evaluate the…

Q: 1. Evaluate (32²-2z) dz, where C' is parametrized by z(t) = t+it², 0<t<1. 2. Evaluate (1 + iy) dz.…

A: Consider the line integral ∫C3z2-2zdz in complex plane, here the curve C is parameterized by…

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: F(1)=r cot"(1.4-r')i +1.4(r² -1) j + Jar +1k t = 0 at and is normal to the plane x+y-z 3

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Q: Q#5. Evaluate the surface integral When f(x, y, z) = 3xyz and g is the portion of the plane × + 3y +…

A: We have to find Surface integral  f(x,y,z) = 3xyz Plane given is x+3y+5z = 6 in first octant

Q: 2. Evaluate the triple in tegral where E is the y-2x2+2z20nd the plane y= 8. solid bounded by SI…

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Q: Evaluate fc F • dr, where F = (z, y2, x) and C is parametrized (in the positive direction) by r(t) =…

A: F=z,y2,x and C is the parametrized by r(t)=t+1,et,t2 for 0&lt;t&lt;2. We have to evaluate ∫CF·dr.

Q: (b) f(x, y) dA where D lies below the paraboloid z = x2 + y?, above the ry-plane, and inside the %3D…

A: Let's find.

Q: 11. If F= (y? +z? - x²) î + (z² + x² – y²)j+(x² + y² – z²) k, evaluate fcurl F-î ds integrated over…

A: Given :-  If  F→ = y2 + z2 - x2 i +  z2 + x2 - y2 j +  x2 + y2 - z2 k To Determine :-  ∫ curl F→. n^…

Q: 2 The values of a and b such that the surfaces ax - byz = (a + 2)x and 4x2y + z3 = 4 cut…

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Q: 8) Given z =f (x.y) =1- (x² +4y2) ,Find : a)The vector function, 10 normal to the surface of f at…

A: This is problem of vector calculus.

Q: Wrons kiant y" + p(+) y # q (4) j'arlt)g =0 +q(4) j' +r (t)y=0 solotions of fifferential equations…

A: Determinant properties

Q: 5 + 2? + y, the cylinder Evaluate e dV where E is enclosed by the paraboloid z = E 2 + y? = 1, and…

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Q: S,F•dr, and C is the triangle, with Use Stokes Theorem to evaluate where F(x,y; =) = (x+ y² )i+ (vy+…

A: Given: Fx,y,z=(x+y2)i+y+z2j+z+x2k and C is the triangle with vertices 1,0,0 ,0,1,0 and 0,0,1 We know…

Q: Let F = < 4x + y², 5y + z, 9z + x So F. dī, Use Stokes' Theorem to evaluate C where C is the…

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Q: The parametric form of the solutions of the PDES -y zz +x zy = 0, :(s, s²) = s³ (s, s*) %3D %3D z…

A: Option (c) is correct

Q: 2. Consider the following oriented surfaces: S1 : z = 0 with downward normal vectors S2 : portion of…

A: Stokes theorem

Q: (ii) Use Stokes' Theorem to evaluate SF · dr, where F(x, y, z) = x²zi+xy²j+z²k and C is the curve of…

A: Given function is F(x,y,z)=x2z i^+xy2 j^ +z2k^ . Plane: x+y+z=1 Cylinder: x2+y2=9 The objective is…

Q: 1. Compute ff,(V × F) - dS, where S is given by r² + y? + (z – 1)? = 2 and z > 0, and F(x, y, z) =…

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Q: Evaluate (2z – z) dz, where C is parametrized by x = -t, y = t² + 2, 0<t< 2.

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Q: 1. Compute ff(V × F) - dS, where S is given by r² + y²+ (z – 1)² = 2 and z > 0, and F(r, Y, z) =…

A: Given: S is given by x2+y2+z-12=2  and z≥0 Fx,y,z=-esinzy,x+sinz,cosx

Q: Evaluate e dV where E is enclosed by the paraboloid z = 6 + x² + y°, the cylinder 2? + y? = 5, and…

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Q: 1. Consider the ellipsoid a?a? + B*y? +v°z? = 7? (a) Parametrize the surface S = r(u, v) by x = and…

A: Given  α=1, β=6, γ=13The ellipsoid α2x2+β2y2+γ2z2=γ2and parametrize the surface S=ru,v…

Q: Use Cartesian coordinates to evaluate JJ v² av where D is the tetrahedron in the first octant…

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Q: JJ F- ndS if F= xi + yj+(z² –1)k Evaluate and o is the curved surface of the cylinder y = V4-x? from…

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Q: Evaluate the surface integral.

A: We need to find the surface integral of

Q: XII. $F•d$ where F = (x,y,z²) and S is sphere of radius 2 with center at 0.

A: F→=x,y,z2, surface S is sphere of radius 2 with centre O=0,0,0To find: ∯SF→.dS→

Q: Find a parametrization x = x(t), y = y(t), tE [0, 1] for the line segment from (-2, –5) to (–3, –7).…

A: To parameterize a  line segment with endpoint ( -2 , -5) and ( -3 , -7 )

Q: A parametrization of the portion of a surface z=x²+y2 below the plane z= 16 is given by 7(u,v)=…

A: Since you have sent more than one question so we can just answer one of them because of our…

Q: Use spherical coordinates to evaluate the triple integral [[[ 2² + 3² + 2²dV, where E is the ball:…

A: We will first change it into spherical coordinate and then solve it.

Q: O 9/ Compute I - [.lryz) , uhere s is the part g the plane y +x that lies inside the half-cylinder…

A: We know that, dS=1+∂z∂x2+∂z∂y2dydx where S is the part of plane z=y+x. ∂z∂x=1∂z∂y=1 dS=1+(1)2+(1)2…

Q: 6. Let F (x,y, z) = (x+ y,–2y + z, 8z – x) and let U be the sphere where x2 + y? + z² = 25 oriented…

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Q: 5) 2z dV, E is the solid enclosed by the cone % = Vr2 + y? and the E plane z 5

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Q: Consider the following. C: y-axis from y = 1 to y = 25 (a) Find a parametrization of the path C.…

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Q: 5. USC cyllndrical coordinates. Evaluatc (k4)dv, where E IS the solid that ies betwern the cylinders…

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Q: Which of the following is the least distance from the origin to the plane 2x + 2y + z = 1? Lütfen…

A: We have to find

Q: 17. F = |r|r, where r = xi + yj + z k, S consists of the hemisphere z = /1 – x² – y2 and the disk x?…

A: The solution are next step

Q: Evaluate (xr + y² + =³)? dV, where 2 is the solid hemisphere (half of the Ω sphere) enclosed by x +y…

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Q: x +y +z dS where S is the part of the plane z = y +4 that is inside of the cylinder x2 +y = 4…

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Q: 1 Let S be the surface parametrized by r(u, v) (u cos v, u sin v, 9- u- -u²), for 0 <u s 3,0 SUS5.…

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