(ii) Evaluate F-dr, where C is any smooth curve from (-1,1) to (1,0) by using Fundamental Theorem of line integral. F(x,y) = (1+ 2xy?)î+ (2x°y +2y)}

Q: Use Green's Theorem to find f. [(2x – y+ 4) dx + (5y+ 3x – 6) dy] around a triangle in the xy plane…

A:

Q: b) Show that F - dr is independent of its path, given that y F(r, y, 2) = 1+. + tan-2) i+ tanrj+ k.…

A:

Q: Calculate the directional derivative of the function f(x, y, z) = 2x³y − ln(y)x²z − z² at the…

A: Answer

Q: (b)Find the derivative of g in the direction of a= 2i+ curl w where at point (0,-1,1) g = In- 4 + 2…

A: Given, w=cosxyzi+xeyz+1j+xyzk Then, Curl…

Q: 6. Set up but do NOT evaluate Evaluate the line integral 2 curve (t) - (t", t), where 0sts0. L ds,…

A: Consider the given integral as ∫Cyxds, where C is the curve r⇀t=t4,t3 and 0≤t≤1. Note that, the line…

Q: Show that if the function f(x, y) = ex siny + ecosx, satisfies Laplace Equation.

A: We can solve this as follows:

Q: In(xy) cos(Tx + T) when x 2 and y = 1/2.- Find the tangent plane to f(x, y)

A: We have to find the equation of  tangent plane.

Q: 21 (important) Find the fixed point for F(x) = ax + s. When is it attracting?

A: Given Fx=ax+s

Q: (b) A bounded, 3B > 0 such that |f(x)| < B Vx € (0,1), continuous function on (0,1) with unbounded…

A:

Q: 8. Find the directional derivative off at the p in the direction of v = () of f(x, y, z)=x²exx, p =…

A:

Q: Find the orthogonal trajectories of the family of curves: y = k (csc x + cotx) Ⓒy² = 2 (C- cos x) O…

A:

Q: a) Evaluate the line integralZ F dr C in terms of t where F = cosyi+xj+ye²>k and C is the line…

A:

Q: (b) Find the derivative of g in the direction of a=2i+curl w at point (0,-1,1) where u u +2 g= In. J…

A:

Q: Prove that the function ((z*)2 Iz| > 0 f (z) = z 0, z = 0 is not analytic at z = 0 by calculating…

A: Given function is f(z)=z*2z, z&gt;00,        z=0 we have to prove, the above function is not…

Q: 2. Let F = 3.xyz i – 4xyz j + 2xyz k . Find div F and curl F. -

A:

Q: (1) (a) Show that the line integral tan(y) dr +x sec (y) dy is independent of path, where C is any…

A:

Q: 3. Find the parametric equation for the line tangent to the curve of intersection of the surfaces x²…

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

Q: Evaluate the line integral F. dr where F = (3 sin x, 3 cos y, 10xz) and C is the path given by r(t)…

A:

Q: A) Show that from the lower relations near the point (0,0,0, 0) u can be written as a derivative…

A: Given relations: vcos(y)+xcos(u)=0 ucos(x)+ycos(v)=0 To find the second order Taylor polynomial of u…

Q: 2 The tangent plane to the graph of the function ƒ : R² → R defined by f(x, y) = −1+ cos(xy) +…

A:

Q: 17. Evaluate the line integral xe" dx +y sin yªdy along the triangular path consisting of the line…

A: Consider the given integral. I=∫Cxex2dx+ysiny2dy The given points are 0,0,2,3,1,0,0,0. Now, draw all…

Q: Evaluate the line integral f. F. dr where F = (2 sin a,–5 cos y, 10xz) and C is the path given by…

A: We will evaluate the given line integral.

Q: Prove that the function (z*)2 1리 > 0 0, z = 0 f (z) = is not analytic at z = 0 by calculating all…

A: Calculate the derivative of fz at z=0.  f'0=limZ→0fz-f0z-0       =limZ→0z*2/z-0z-0       =limZ→0z*z2

Q: 4. Curl of f (x, Y, z) = 2xyi + (x² + z²)j + 2zyk is

A:

Q: (3) Consider the derivative dz = 2xydx + (x² + 2y)dy Evaluate whether this is a total derivative, by…

A: We will check whether the given function is total derivative or not. We also integrate dz pathwise.

Q: tion z2 – (x2 + y2)2 and let p = (0, 0, 0). ermine the set Tp(Zf ) of all tangent prs to Zf at p. Is…

A: Sol

Q: 2. Integrate f(x, y) = sin /x2 + y² А. over the closed unit disk.

A:

Q: a) Plot a direction field and some integral curves  for xy'-2y=-1 on the rectangular region {-1<or…

A: a) given differential equation  xy'-2y=-1 claim- plot a direction field and sole integral curves for…

Q: 6. Find an equation of the tangent plane and a set of symmetric equations of the normal line to the…

A: Let's find.

Q: Find the orthogonal trajectories of the family of curves: y = k (csc x + cotx) Ⓒy²= 2 (C- cos x) O y…

A: This question is related to differential equations, we will solve it using given information.

Q: Let (x, y) = 3y² - 2y – 3x² + 6xy. a. Find the equation of the tangent plane at the point (1, -1).…

A:

Q: The direchional derivative of the salan funchion at (l,-1,0) in the direction (i-j+2k) is xy- z- xyz…

A: Given that  Let f(x,y,z)=x^2y-y^2z-xyz To find the directional derivative at point

Q: 2. Find an equation for the plane tangent to the surface 3x2 - 4xy+ z2 = 0 at the point (1, 1, 1).…

A:

Q: Evaluate the line integral F. dr where F = (3 sin x, 3 cos y, 10xz) and C is the path given by r(t)…

A:

Q: Compute F. dr where F (x, y) : (7xy + x cos x) i+ (-yev + 6x²) j and C is a piecewise closed curve…

A:

Q: B) Evaluate the line integral F-dr, where F (x, y) = x²y³i+ x°y°j and C is the curve r (t) = (t³ –…

A:

Q: 3. Let f(x, y) = xy² + x°y². Find an equation for the tangent plane at (x, y) (-1,2). %3D

A:

Q: 2. Let closed curve C consist of the line segment from (0, 0) to (2, 1), followed by the line…

A: We will solve both parts of this question using basic knowledge of line integrals and definite…

Q: Let C denote the curve of intersection between x = -5z° and y = 6z", starting from A(0,0,0) and…

A:

Q: find div and curl F F(x,y,z)= 7y3z2i- 8x2z5j-3xy4k F(x,y,z) = exyi-cosy j +sin2 z k

A:

Q: Tangent to the curve z2 2xy + x² , z = 3 , at the point ( 1, 4 , 3 )

A: Consider the curve, z2=2xy+x2 ⇒x2+2xy-z2=0

Q: (1) Show that F(x, y) : (2x + y, x - y) is path independent and use the fundamental = theorem of…

A:

Q: Tangent to the curve z2 = 2xy + x² , z = 3 , at the point ( 1, 4 , 3

A:

Q: prove that x^2y/(x^2+y^2) is not differentiable at 0,0 when  x^2y/(x^2+y^2) if (x,y) not equal to…

A: f(h,k)-f(0,0) =hfx(0,0) +kfy(0,0) +h2+k2∅(h,k) h2kh2+k2-0 =hA+kB+h2+k2ϕh,k ...(1) A =fx(0,0)…

Q: /alue of divergence of function (x2y2)î - (z³y – 4x) ĵ + 4y2 xk at (1, 2, 1) is

A: Let f = (x2y2)i^ - (z3y-4x)j^ + 4y2xk^

Q: (2) Find an equation for the tangent plane to the graph of z = where z = f(x, y) is implicitly…

A: Given: The function is fx,y,z=xy+xz+yz-lnz2+1-4. Calculation: Obtain thethe value of  fxx,y, fyx,y…

Q: (a) Calculate the line integral of F =r°yi + y°j + z²k along the straight line path from (1, 1,0) to…

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Q: how that the function satisfy Laplace equation f(x,y) = In /9xy+3y²

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Q: Let F(r, y, z) = (y² + 2z, 2ry + 4, 2a), and let the curve C be described by x = t, y = t, z = t3…

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Q: 4. f(z) = { |z|* z = 0 verify that the Cauchy-Riemann equations are satisfied at the origin z =(0,0)…

A: