Show that lim (z,y)→(0,0) 4 z - Va² + y? V16 – x² – y? – 2²

Q: Evaluate Se (5xy + x² + y²) dx + (x² - y)dy by using Green's theorem, where C is a closed curve that…

A:

Q: Let f = zy + yx, v = [y, z, 4z - x], w Find - [vª, zª, x®]. (a) grad f and f grad f at (3, 4, 0) (b)…

A:

Q: Show that each function in satisfies a Laplaceequation. ƒ(x, y, z) = 2z3 - 3(x2 + y2)z

A:

Q: lim sin (2 – t) cot² (t² – t – 2) t+2

A: We are given: limt→2sin2(2-t) cos2(t2-t-2) We need to evaluate the given limit This can be evaluated…

Q: (a) Find the solution of the limit using polar coordinate system. Vx? + y? x² + y? lim(xy)-(0,0)

A: To find:lim(x,y)→(0,0)x2+y2x2+y2 Take x=rcosθ, y=rsinθ to convert the term from cartesian to polar.…

Q: If f(r, y, z) does lim f(r, y, z) exist? %3D 1² + 2y? + z² ' (z,y,z)¬(0,0,0)

A: To find its limit we have to first convert the function into polar form.

Q: Let F(x, y, z) = 6x?i – cos(xy)(i + 5). Calulate the divergence: div F(x, y, z) =

A:

Q: For F(x, y, z) = x²z² i+ sinx z²j+zx²y*k, find div F and curl F.

A:

Q: Prove that flx)= (i+x)? + X and diverge Litx)? both

A:

Q: Find the arclength of the graph of the fun ction over the indicated inter val. (a) y = 2x3/2 + 3, 0…

A: Given : Functions and their interval (a) y = 2x3/2 + 3 , 0≤x≤9 (b) y = ln(cosx), [0, π3] (c) x =…

Q: x'ey Let f(x, y) : Then lim (xy)-(0,0) x2+3y2 f(x,y) (A) exists and equals 0 since f (x,y)…

A:

Q: Numerically integrate f(x) = x2 from a=0 to b= 2 using a multiple segment (n-2) trapezoidal %3D…

A: Trapezoidal rule  ∫abf(x)dx≈Δx2​(f(x0​)+2f(x1​)+2f(x2)+2f(x3​)+⋯+2f(xn-1​)+f(xn​))whereΔx=b-an

Q: Exercise 11. iz Let f(z) and CR be the upper half circle with center (0,0) and radius R z5+1 taken…

A: Given problem is :

Q: (a) Find the solution of the limit using polar coordinate system. x² + y2 lim (x,y)-(0,0) x2 + y?…

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

Q: Evaluate the limit: r(t + h) – r(t) lim h >0 for r(t) = (t°, sint, 1) h r'(t) = (

A:

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

A:

Q: find the divergence of F. F(x, y, z) = x²i + y²j+z²k div F = 2(x + y+ z) div F = 3yz? + 2y sin z +…

A: Topic = Derivative

Q: 35. f(x, y) = x+2y2 -x; R is the disk x2+ y? < 4.

A:

Q: Utt Urr 100, x > 0, t > 0. u (x, 0) = u (x, 0) = u (0, t) = 0 lim u (x, t) = 0, x → 00

A: 1. Laplace transform is an integral transform that converts a function of real variable t to a…

Q: Use polar coordinates to find the limit. [Hint: Let x = r cos(0) and y = r sin(0), and note that (x,…

A:

Q: ind the curl and divergence of F(x, y, z) = (e², –2ev², 3xe").

A:

Q: Q2. If F = (x² − y² + 2xz)ī − (2xz − xy + yz)j + (z² + x²)k, find the of curl F and curl of curl F…

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: Use polar coordinates to find the limit. [Hint: Let x = r cos and y = r sin , and note that (x, y)…

A: To find limit: limx,y→0,0x2y2x2+y2 Solution: In polar coordinates we have, x=rcosθ and y=rsinθ  And…

Q: A surface x(y, z) is defined implicitly xy + yz +x'z' = 3-xZ Calculate ду

A:

Q: Use polar coordinates to find the limit. [Hint: Let x = r cos(0) and y = r sin(0), and note that (x,…

A:

Q: Evaluate. where G is a solid that lies in the first octant bounded by x^2 + y^2 = 3z and z = 3.

A:

Q: Find div F and curl F if F(x, y, z) = 4y³z'i – 11a"z°j – 4xy°k. - -

A:

Q: (a) Find the solution of the limit using polar coordinate system. x² + y² x2 + y2 lim (x,y)→(0,0)

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

Q: Find the domain and limit ast approaches infinity of T(t): i+ [2e ]j+[arctan(t)] k 3 In %3D 7.

A: Given : r(t)=lnt+1t−3i+[2e−t]j+[arctan(t)]k

Q: Let C be the boundary curve counterclockwise of the semidisk x² + y? < 9, y 2 0. Then f.(-2ydx +…

A: A detailed solution is given below.

Q: Use polar coordinates to find the limit. [Hint: Let x = r cos(0) and y = r sin(0), and note that (x,…

A:

Q: -) Let -y M(x, y) = x2 and N(x, y) = + y? x2 + y? how that ƏN ƏM dx ду ) Show that -y F(x, y) = x² +…

A: Given :-  A vector field for conservative field in the plane ,  ∂N∂x = ∂M∂y To Determine :-  M x,y =…

Q: 川- Q1 Verify that [(y² – 2x )dx +(2xy +x² )dy = [[] dA where C is the boundary of a regi ôy R R…

A:

Q: ry lim (z.v)¬(0,0) + y² exists.

A: Please refer the attached image for complete solution.

Q: Find fæ, fy, and fz. f(z, y, z) = z In(a"y cos(5z) 4.3 = z In (x*y° cos(52)) fr = fy = f: =

A:

Q: Let C be the boundary curve counterclockwise of the semidisk x2 + y? < 9,y 2 0. Then f (-2ydx +…

A: Given Information -  C is the boundary curve counterclockwise of the semi disk x2 + y2 ≤ 9 , y ≥ 0…

Q: Find the curl and also the divergence of F(x, y, z) = (x²z, e" + zz, xyz)

A: The objective is to find the curl and divergence of Fx,y,z=⟨x2z,ey+xz,xyz⟩ Curl and divergence of…

Q: b) Use the divergence theorem to compute ff, F · i ds, where S is bounded by x +y + 2z = 2 located…

A:

Q: H.W: Select the divergence of F(x, y) i+ (2x – 3y)j. %3D 1 (b) -* +2, y2 1 (c) (a) 3, (d) -2. - - -

A: Multiple:   Divergence of a vector field find by taking the scalar product of the vector operator  ∇…

Q: Evaluate the limit: r(t + h) – r(t) lim h→0 for r(t) = (1-6, sin t, 8) h r'(1) = ( ...

A: We have to find components of r't

Q: evaluate both sides of divergence theorem for A= ra,+rcos(8)a , +r²a, through the closed surface…

A: Given problem is :

Q: Let YR(t) = Reit for 0 <t< 1. Prove that %3D 1 dz = 0. lim R00 JYR z3 + 2z You may use Proposition…

A:

Q: %3D a) Find fyxyx for f(x,y) = In(xy) – cos (xy) at (1,t).

A:

Q: se the Divergence Theorem to evaluate (8x + 12y + z2) ds Is here S is the sphere x2 + y2 + z² = 1.

A: To compute the below integral using divergence theorem.

Q: Find div F and curl F if F(x, y, z) = 9y%z®i – 4x°z1°j – 5xy°k. - -

A:

Q: Find the analytic functiopn f (z) = u (x,y) + i ( x, y ) such that u ( x, y)=  4xy3-4x3y+x.

A: