Verify the Divergence Theorem SI div F av F. dS S E for F(r,y,2) = z i+ yj+xk and where E is the solid ball r? + y? + 2² < 16.

Q: Evaluate f. F.dr, where F(r, y, z) = z?i+rj and C is the boundary of the square {(r, y, z)|0 < x <…

A:

Q: Evaluate ſç(2x + z², − 3y + x, 6z − 4y) · ds where S is a sphere of radius R=3 and center the axes…

A:

Q: Q2/ Find T, N, and k for the space curve r(t) = (3 sin t)i +(3cos t)j + 4tk

A: Question: Objective is to calculate the tangent vector T, normal vector N and the curvature κ of…

Q: Let a differentiable function g and f(x,y) = g(x²y²), g(4) = 4, g'(4) = 1 be given. Then, find the…

A:

Q: Find the f(z) = u(x, y) + j v(x, y) for which X b) u = e cosy most general analytic function

A:

Q: Verify Stoke's theorem for the vector point function yz2 (xz² .3 + ex over the surface S of the cube…

A:

Q: Assume that u and u are continuously differentiable functions. Using Green's theorem, prove that Uz…

A:

Q: Gixen dw wt 3y x dt dy +2X +3 x Find the gencral oludtii, usri eign.

A: Write the given system into matrix equation as follows.

Q: 3. [32] −1 4-3 4. 5. [. |_3 -4 2 1 2 -1 4,

A:

Q: he upward flux of

A: Introduction: Let F be a function and let the region is S, then the flux of the function is given…

Q: (u, v) ~ (x, y) as (x, y) → (0,0) that will bring to the sum of squares z = A1u? + A2v² the…

A: given u=ux,yv=vx,y   and u,v~x,y as x,y→0,0 define a function z=λ1u2+λ2v2 where…

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: Find the maximum value of f(x, y) = 11zy(9 – 99x – 9y) over the triangle with vertices (0, 0). (†,0)…

A: Given: The triangle vertices are (0,0),(111,0) and (0,1) f(x,y)=11xy(9-99x-9y)…

Q: If v (x, y, z) = 6i-12j+7k find 1) Divergence of v. 2) curl of v.

A: Solution :

Q: af Let f(x, y, z) = xy + z². Calculate where x = = s², y = 2st, z = t². მs

A: Step 1: Step 2:

Q: SF - dr si F(x,y,2) = xy? i + (x² + y²)j+ 2yz k y r(t) = vE i + t² j + t k, 1sts4.

A: The solution are next step

Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, 2) = (2x²y² + In z) i – xy³ j – tanh(a² + y³)…

A:

Q: c) Given a function $(x, y, 2) = 3x² + In (3y² + 1) + e* cos z. i. Find the directional derivative…

A:

Q: F. dS, where F(x, y, z) = z²xi + 2) + (x2z + y2)k and S is the top half of the sphere x² + y² + z? =…

A:

Q: Q3A) Find normal line and parallel plane to the function f(x, y, z) = xz siny-√3 At point (1,1,2).…

A:

Q: Q5. Analyze and Verify Gauss divergence theorem for F 2 xz i+ yzj+ z? k over the upper half of the…

A: Here, F=2xzi+yzj+z2k Given upper half of the sphere x2+y2+z2=a2. Let ∅=x2+y2+z2-a2=0…

Q: ôf ôf and - for the fiunction f(x, y,z) = x²y+y° + xz? ôx 'ôy Q2: Find .2

A: We have to Solve the partially differential .

Q: Let F(x,y,z) = (y,x²,zy) where C:R(t) = (cost,sin t , 3t) and t e [0,2]. Evaluate %3D %3D •dR

A: We have to find value of given Integral.

Q: Calculate fff, V.F dV where V is a closed space bounded by x = 0, x = 1, y = 0, y = 1, z = 0 z = 1…

A: In general, triple integrals involve the integral with respect to x, y, and z. Similar to partial…

Q: a) f(x, y, z) = 0 b) f(x, y, z) = 1 c) f(x, y, z) = (y + 2, z)

A:

Q: Consider the 2D domain D = (x, y)|y > 0, ; < x² + y² < x 1. 1 d(x, y) using / / Calculate dr d0 set…

A: