4. let f(x, y) = x² – xy + y² – y. find the direction u and Duf at Po(1,–1) for which Duf is the largest.

Q: 1. Use an antiderivative to show that for every contour C extending from a point z₁ to a point 22, 1…

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Q: Suppose w = In(x² + y² + z²) , x = ue³" sin(9v) , y = ue3" cos(9v), and z = ued". %3D Compute dw/du…

A: If w is a function of three variables, such that w(x,y,z) and x, y and z are further function of u…

Q: Consider the functions z =2e* In y, x In (u cos v), and y= u sin v. dz (a) Express and du dz as…

A: Since you have posted a question with multiple sub-parts, we will solve first two subparts for you.…

Q: Q3:Let w = In(x² + y2 +z2) %3D x = ue" sin u y = ue' cos u %3D z = ue" Find : aw A) by chain rule du…

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Q: 2) Use Green's Theorem to compute f F df where F= (3x +4y)i + 2xyj and C is the triangular path from…

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Q: Verify that u is a solution to the equation ugx + Uyy + u zz = 0. u(x, y, z) = e4z+3y cos(5z)

A: Follow the procedure given below.

Q: 4x Find the linearization L(x,y) of the function f(x,y) = e cos (3y) at the points (0,0) and0,

A: Answer is given below

Q: Let g(r, y) = ye-, r(t) = In(t² + 1), and y(t) = vt. Find g(r(t), y(t)).

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

Q: Demonstrate that the functions Y1 = a", y2 = sin(x), Y3 = xe* are linearly independent on R. 9.

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Q: Q1 If f(x,y, z) = 2 x²y + ez+ Vz Inx %3D Find fr fy and f2

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Q: 30. (2x + y²) dx + (2xy + 3y) dy C C: Any simple closed curve in the plane for which Green's Theo-…

A: We know that Green's theorem

Q: Consider the functions z=7 ex In y, x= In (u cos v), and y= u sin v. dz dz (a) Express and as…

A: Given, z= 7 ex ln y    ....... (1)x= ln ( u cosv) .......... (2)y= u sinv…

Q: Let flxiy) = Rx Jy 27 %3D a) Finid the gradrent of f b) Find the directional derivative of f at the…

A: Consider the function as  fx,y=2xy

Q: 3. Let X(t) = t2W (t) – 2 f sW(s)ds, Is X(t) a martingale?

A: X(t)=t2W(t)-2∫0tsW(s)dsWe know that, X(t) is a martingale, if,EX(t+1)=X(t)

Q: find the linearization L(x, y) of the function ateach point. ƒ(x, y) = x3y4 at a. (1, 1), b. (0, 0)

A: Consider the function: fx,y=x3y4 To calculate the derivatives of the given function with respect to…

Q: Find the linearization L(x,y) of the function f(x,y) = e (X cos (5y) at the points (0,0) and 0,- 2

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Q: 2) Use Green's Theorem to compute f F. df where F=(3x +4y) i + 2xyj and C is the triangular path…

A: Use green's theorem

Q: 2) Use Green's Theorem to compute f F df where F=(3x + 4y)i + 2xyj and C is the triangular path from…

A:

Q: 4. Let f(z) = 1 23 1 - and find [ f(z) dz, where C a circle of radius 2 about the origin with…

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Q: 2. Let F(t) = (e' cos t) î+ (e' sin t)j+ek. a. Find symmetric equations of the line tangent to the…

A: Given:  F→t = et costi^ +et sint j^ +et k^  To Find:  Symmetric equations of the line tangent to the…

Q: Express the curve by an equation in x and y given x(t) = sin(t) and y(t) = 8 + cos² (t). a) O-x² +y…

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Q: Compute I = (2x²132²y+ 4xy)ax +(x'+2x²-y') dy+zdz along The curve C C4U Cz UG given in the figure…

A: Given To compute I=∫C2x3+3x2y+4xy.dx+x3+2x2-y2dy+zdz along the curve C=C1∪C2∪C3 given in the figure.

Q: Let Dy be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function.…

A: The linear transformation Dx:C'a,b→Ca,b maps a function fx to its derivative Dxf. Here, given that…

Q: Let F(x, y) = G , y 1 )- Calculate the work against F required to move an object from (1, 1) to (3,…

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Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = (sin xy)/z at…

A: Linearization of a function of three variable[say f(x,y,z)] at (a,b,c) is…

Q: Evaluate O F.dT where +(3y %3D y + 1 and C is the triangle with vertices: (0,0), (6,0), (3, 5).

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Q: xy ) Suppose thot fixy)=e , x(4,0)= 3uSiina, glu, vl = f(x (4,v), yluiv), find , by using Chain Rule

A: Using the chain rule,

Q: Compute the discriminant D(x, y) of the function. S(x, y) = x' + y* – 6x – 2y + 1

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Q: 15. Evaluate the line integral J. (3x²y* - 6xy ]dx + (4x²y³ - 3x°) dv, where C is any path from…

A: My=Nx means F is conservative vector field

Q: 2. Let F(x,y) = (cos(x – 2y), –2 cos(x – 2y)). Find curves C and C2 so | that, F. dr = 0 F. dr = 1

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Q: 9. Demonstrate that the functions Y1 = x2, y2 = sin(x), y3 = xe are linearly independent on R.

A: Recall: If y1,y2 and y3 are the given function and         Wy1,y2,y3x = y1y2y3y1'y2'y3'y1''y2''y3''…

Q: | Let u = x*y + y°23 where x = rse', y = rs²e=t, = 2, 8 = 1 and t = 0, by using Chain Rule. p2 Sint.…

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Q: Find aw in terms u and v if du w= In(x2+y2 +z²), x = ue" sin u, y = ue' cos u, z = ue"; (u, v) =…

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Q: (b) If F = (5xy- 6x )i + (2y– 4x)j, Evaluate [F.dr along the curve C in the xy-plane, y =x' from the…

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Q: 2. (a) Let z ry, r =r cost and y=r sin t. Compute by using the chain rule and express your answer in…

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Q: Assume x and y are functions of t. Evaluate dy for 4xy - 7x+4y3 = - 48, with the conditions dt dx =…

A: Explained below

Q: Use Green's theorem to evaluate S,(5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A: The posted question has multiple questions but according to guidelines we will solve the first…

Q: (ii)If y=(x+Jx* +1. prove that y'= Vx +1

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Q: Evaluate § F · ds, where F = 2x(x+y)i + (x² + xy + y?)j and the positively-oriented curve C encloses…

A: Here we will use greens theorem to find line integral of vector field For a vector field F→= Pi^+Qj^…

Q: ) Find the direction in which f (x, y) = x²+xyincreases most rapidly at P(1, –1).

A: 1.) given multivariable function is, fx,y=x2+xy and point is P(1,-1). Divergence: Divergence of the…

Q: 2. If F(x, y, z) = y²i+(2xy+e3=)j+3ye³=k. Evaluate F·Î ds, where C is the curve represented by r(t)…

A: Given F→x,y,z=y2i^+2xy+e3zj^+3ye3zk^ and the curve is C given by r^t=cost,t,sint,  0≤t≤π2. Find…

Q: 7. Let F = (2x sin(y)e", x² cos(y)e*, z² sin(y)e*). Find along the path c:T = (sin(t), t, cos(t))…

A: F=2xsinyez, x2cosyez, x2sinyez curlF=ijk∂∂x∂∂y∂∂z2xysinyezx2cosyezx2sinyez⇒curlF=0 So, F is a…

Q: Show that F(x, y, z) = (y² – e* sin z)i + (2xy + z)j + (y – e* cos z)k is conservative and hence…

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Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = tan-1 (xyz) at…

A: Consider a function: fx, y, z=tan-1xyz……1   Given point, say A, is: A=1, 1, 1   On comparing A with…

Q: Show that the function u = 9(ry) + /TY v(y/x) provides the uation ‚²u y? 0. dy?

A: First find the partial derivative of the given function with  respect to x :

Q: x + Y f(x, y, z) = x sin + e*% – 1

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Q: Use Green's theorem to evaluate f (x²y) dx + y²dy here C is close path formed y = x and y = x³ from…

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Q: 2. Suppose that a function f: R3 → R³ defined by: f (х, у, 2) 3 (2х + Зу — z, sin(тхy?z-1),…

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