3. Find the linearization L(r, y, z) of the function f(x, y, z) = y² sin(2rz) at Po = (7,2, 4). Then find an upper bound for the magnitude of the error E in the approximation f(x, Y, z) z L(x, y, z) over the region R. R: |r – a|< 0.2, |y– 2| < 0.3, |z – |<0.1

Q: (d) |// Vx² + y² dV, where P is the region under the cone z = 2 – ry-plane. /x² + y² and above the…

A: Since you have asked for a multiple question, we will solve the first question for you. If you want…

Q: Find the value of the line integral  ∮ C (2*y + e^x) dx + (ln(y) + 8*x) dy, where C is a closed path…

A: Let F=Px,yi+Qx,yj be the vector field and C be the boundary of the region D. The Green's theorem…

Q: 12. In the xy plane the region D is bounded by the lines y = 2x + 6 y = 2r +1, y = -3r +6, and y =…

A: Given: y=2x+6 y=2x+1 y=-3x+6 y=-3x+1

Q: (6) Use the Divergence Theorem to evaluate dS, where F(x, y, 2) = xyi + (y² + e*=*)j + sin(xy)k and…

A: Given,         and

Q: B) Integrate f over the given region 1. Square f(x, y) = 1/(xy) over the square 1 ≤x≤ 2, 1≤y≤2 2.…

A: In each part we need to find ∫∫R f(x, y) dx dy where R is the given region

Q: Evaluate E=∭D(x2+y2+z2)−3/2dV where D is the region bounded by the surfaces x2+y2+z2=9 and…

A:

Q: Q.1B) Prove that: I = f, Cos () dxdy sin(1) %3D 2 where R the region bounded by x+y 1,x = 0,y = 0

A:

Q: Find the linearization L(r, y, z) of the function f(r, y, 2) = y² sin(2r2) at Po = (r, 2, 4). Then…

A:

Q: A flux is given by F = x²y³ i – xy j. Verify the Green's theorem for the region bounded by the…

A:

Q: 10. Use Green's Theorem to evaluate the line integral (e" - y) dx + x cos(y²) dy, where C is the…

A: 1/2 + ( (1/2 )sin 1)

Q: Let f(z) = sin z. Use the Maximum Modulus Principle to find the maximum value of f(z)| as %3D z…

A:

Q: (c) By using the Green's theorem, evaluate $(2xy) dx + (4x²y*) dy if C is the boundary of a region…

A:

Q: 9. Evaluate 18x dV where E' is the region behind the surface y = 4 - 2² that is in front of the…

A:

Q: Use the Divergence Theorem to evaluate ff F-n do where S F(x, y, z)=(x² +cos yz, y-e², z² + x²) and…

A:

Q: 6- Evaluate the line integral ScF · dr, where F = [xy, yz, xz] and C is the boundary of the…

A:

Q: 12. = y dV, where E is bounded by the planes x = 0, y = 0, z = 0, and 2x + 2y + z = 4

A: We need to find ∫∫∫Ey dV E is bounded by the planes x=0, y=0, z=0, 2x+2y+z=4 From 2x+2y+z=4 we get:…

Q: Calculate the value of the line integral fds where f(x, y, z)=e² and C is described by X C (x, y,…

A: The objective of the question is evaluate the definite integral.

Q: 1. Compute fSSp(z+1)°dV where E is the region lying inside the sphere x² +y² + 22 = 1 of radius 1…

A:

Q: Verify the Green's theorem for integral x²y dx + xy dy, where C is the boundary described…

A:

Q: (15) Use the divergence theorem to evaluate SSF. ds, where S 3 F(x, y, z) = and 5 is the surface of…

A: To evaluate the given integral using divergence theorem.

Q: 2) Evaluate the following line integral:S. (v + z)dx + (x + z)dy + (x + y)dz where C consists of the…

A:

Q: Let f(x, y) = 3x² + 2y² − 2xy — 1. Find the local maximum and minimum values and saddle points of f,…

A: Double integral, Maxima minima

Q: Verify the divergence theorem for F=3i + xy j + x k  taken over the region bounded by z = 4 - y^2 ,…

A:

Q: 8. Use Green's Theorem to calculate f, x?y³dx + (xy - y?)dy where C is the boundary of the region…

A: The given integral is ∫Cx2y3dx+xy-y2dy, where C is the boundary of the region lying between the…

Q: Evaluate the triple integral. 2xy dV, where E lies under the plane z = 1 + x + y and above the…

A: Final answer is 65/84.

Q: 22. Use the divergence theorem to set up only a triple iterated integral for the flux ff, F.ds,…

A:

Q: 1. Let D be the region bounded by the lines z+ y = 0, z + y = 6, z = 2y, and z = 2y – 3, and let…

A: Integrate according to the variables x,y. Find the intersection of the area bounded by all the lines…

Q: Evaluate . (y² z)dV where E be the region in the first octant E bounded by the planes x = z, x + z =…

A:

Q: 3. Evaluate /// 2* dV if G is a solid in the first octant bounded by the plane y + z = 2 and the…

A: triple integral

Q: Evaluate the triple integral. SSSE 7xy dV¸ where E lies under the plane = 1+x+ y and above the…

A: We have to Evaluate the triple integral.

Q: |Evaluate the line integral: f.(r² + y) dx + (4x – cos(y)) dy where C is the boundary of the square…

A: According to Greens theorem, the line integral ∮CPdx+Qdy=∬DQx-PydA. BY the extended version of…

Q: Use Green's Theorem to evaluate |(e* +y²) dr + (+r²) dy where C is the boundary of the region…

A:

Q: Find the linearization L(r. y, 2) of the function f(r, y. 2) = y² sin(2rz) at Po = (7,2.4). Then…

A:

Q: Find the maximum and minimum of w(x, z) = 2xz + z^2 +1 in a region R bounded by x = 0, z = 0, and x…

A: The bounded region R is:

Q: 6- Evaluate the line integral fF dr, where F = [xy, yz, xz] and C is the boundary of the hemisphere…

A: the line integral f F dr, where F = [ry, yz, xz] and C is the boundary of the hemisphere x² + y² +…

Q: 3. Let g(x,y) = Vy+1 In(x + y ) . 5 (a) Find the region D in R which is the domain of g. (b) Make a…

A:

Q: 5. Evaluate z dV, where E is bounded by the cylinder y? + z2 = 9 and the planes x = 0, y = 3x, and z…

A:

Q: Find the area of the specified region. sec e Inside the dircle r 3 and to the right of the line r =…

A:

Q: Use the Divergence Theorem to evaluate |F.n do where F(x,y,z)=(x² +cos yz, y – e" ,z² +x*) and S is…

A: According to question given :F(x,y,z)=x2+cos yz, y-ez, z2+x2S is the boundary of the solidBounded by…

Q: Verify Green’s Theorem for F⃗ = , where C is the boundary of the region bounded by x = 0 , y = 0 ,…

A: Given that F=3x2-8y2,4y-6xy bounded by x=0,y=0,x+y=1Verify green's theoremGreens theorem is∫C…

Q: 2. Let R be the region in the first quadrant of the xy-plane bounded by the hyperbolas xy = 1, xy =…

A:

Q: 7. Use the divergence theorem to find the outward flux of F= 3xy²i+ xe² j+ z° k across the boundary…

A:

Q: Let D be the planar region bounded by y = 0, y = x², and x = 1. The integral ff cos(z² + y²) dA…

A:

Q: 17. Let R be the region between the graphs of y = 1 + e* and y = 1-e* on [0, In 2]. Find the center…

A: Here, f(x)= 1+ex g(x)=1-e-x a=0 and b=ln2

Q: 17. 6xy dV, where E lies under the plane z = 1 + x + y and above the region in the xy-plane bounded…

A:

Q: 3. Use Greens theorem to evaluate (e* + y²)dx + (e' + x²) dy where c is the boundary of the region…

A:

Q: A. Evaluate and sketch the line integral :1 = f yx dx + (2X – Y)dy around the region bounded by…

A: Greens theorem

Q: Evaluate ſ, z dV where E is bounded by the plane x + y = 2 and the cylinder y? +z² = 1 in the first…

A: Given that, ∫∫∫EzdV, where E is bounded by the plane x+y=2 and the cylinder y2+z2=1 in the first…

Q: Calculate the value of the line integral fds where f(x,y,z)=e² and C is described by C (x, y,…

A:

Q: Find the absolute maximum and minimum Of the function f(x, y) = x + Y on the elliptic region {(x, y)…

A: