Let f(x,y,z) =x +y? + z, and a= 2i + j+ 2k then af in direction a at the point да (1,1,1) is А З В 4 C 7 D None of these.

Q: Use Green's theorem to evaluate Sc(y² + x³) dx +xª dy where C is the perimeter of the square [0, 1]…

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Q: Find ôz|ðu and ôzlôv for z=4e* In(y), x=ln(ucos(v)), y=usin(v) at the point (u, v) = (2, 7 | 4).

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Q: Let L(z, y) f(1, y) at (1, 1) . Then f-(1,1) = = y-2x +3 be the local linear approximation to O a. 3…

A: Topic:- partial differentiation

Q: Find the work done by the force F (r, y) = yxi– (x – y) j over the straight line from (1, 1) to…

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Q: Let L(x, y) = y- x+ 3 be the local linear approximation to f(æ, y) at (1, 1) .Then f(1, 1)%3D a. 3.…

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Q: Find the work done by the force F = xyi + (y - x)j over the straight line from (1,-3) to (3,-9).

A: Given force is F→=xyi^+y-xj^ and straight line coordinates are 1,-3 and 3,-9

Q: In(u cos v) and y = usin v. Evaluate and z at the point 2е" In y, x 3D (u, v) = (2, 7/4). Let z = du…

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Q: Compute the directional deri vative of the function f(z,y, 2) = zy² + y²9+°% at the point P(4, –2,…

A: Given function is fx,y,z=xy2+y2z3+z3x

Q: The work done by the force F = in the displacement along the path C:y=x² from the point (0,0) to…

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Q: Utt = 8uxx for 0 < x < ™ u(x, 0) = x² u(x, 0) = 1/2cos(8x) Uz(0, t) = u„(T, t) = 0 %3D II

A: The given P.D.E is                            utt=8uxx for…

Q: Find the work done by the force F = x²yi– xy²j along the paths shown from (1, 1) to (4, 2). (4, 2) a…

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Q: = a + zy", z - w* + w, y I = uu" + w", y u+ ve fi az Find when u = 1, v 2. w=0 = %3D H.

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Q: 2. Find the directional drevative of f(x, y,z) xsinyz at the point (1,3,0) in the direction of v = i…

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Q: 3. Show that F(x,y, z) = (1+ yz) i+ (z + xz)j+ (y + xy) k is conservative. Hence find its potential…

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Q: Integrate f(x,y,z) = x+ Vy -z° over the path from (0,0,0) to (3,9,3) given by C,: r(t) =t i+tj,…

A: The given function is, Let, Also the given paths are, So,

Q: Find Vf at the given point. f(x,y,z) = x3 + y2 – 3z2 +z In x, (1,2,4) Vil(1,2,4) =Di+j + ()k ( %3D

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Q: Find over xlapping area two equatione also gemetrical nepresentahon 2. 4'+v?=n, u?+cV- vn)=1 whore n…

A: Given curves are,…

Q: If f(x, y, z) = In(x7 + y +2)+V + y + z), then find fr, fy and fz at (1, 1, 1). %3D

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Q: Find a potential function for the vector ficld F(x, y, 2) = 3x²i j| 2z In yk. %3| A) r3 + 2z ln y +1…

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Q: af Calculate at point (-1,2) where f (x, y) = 8+ x² + 3y? dv and v = i+j. 10 2 5/2 V2

A: fx, y=8+x2+3y2  &amp;  v=i+j Partially differentiating with respect to x, we get fxx, y=0+2x+0=2x…

Q: 1. Consider f(x, y) = 2x² – y² + 3x – y and U is any unit vector in R2. Find the maximum value of…

A: To find maximum value of DUf at x=1, y= -2 we shall use the given definition

Q: Integrate counterclockwise z + 6 dz, C: |z – 1|= 4 Z – 2 z + 6 dz 2 ||

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Q: Find a non-zero and non-constant vector function F(x, y) = F1(x, y)i+ F2(x, y)j, ƏF OF 02F and dy?…

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Q: If vector T = (x+ y+1)i+ j– (x +y)k then T.curl(T )is

A: We are given vector T.

Q: Find the work done by the force F= xyi + (y - x)j over the straight line from (1,2) to (3,6).

A: we have to find the work done by the given force

Q: a) Evaluate the line integral S. (2x – y) dx – (x + 3y) dy, where C is a straight line from (1,1) to…

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Q: Find the linearization of f (x, y, z) = xy / z centered at (2, 1, 2). Use itto estimate f(2.05, 0.9,…

A: Our Aim is to find the linearization of f(x,y,z) =xyz centred at (2,1,2).and then using that value…

Q: Find the work done by the force F = xy i + ( y - x)j over the straight line from (1, 1) to (2, 3).

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Q: Evaluate fz² In z dz using integration by parts O² Inz-+c Onz-+c Onz-+c 2²lnz - 2 + c 12 21

A: Solution:  step-1: Evaluate ∫z2*lnz dz using integration by parts                Formula for…

Q: 1. (a) Let /(r,y)- cos(z + 2y). Compte the directional derivative Da/(r.y), where u- (b) Compute ff,…

A: Given that, In part (a), f(x, y) = cos(x+2y) u=(x,y)=-12,12 We have to find the directional…

Q: If an object moves along the line segment 'c' given by x = t, y = t?, t3 from P(0, 0, 0) to Q(2, 4,…

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Q: Let F(x, y, z) = (8x7 In(7y² + 4) + 6z') i + ( +4 14yx + 4z)j + (42xz°+ 4y – 27 sin 7z) k and let…

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Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann equations if ux = v, and u,…

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Q: Find the dircctional deriu atrue Du f(-3,-1,3) for the function f Cxiy,2) =xy + yz +ZX in the…

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Q: 4. If ricost cosh t+j sint sinh t, where i and j are constant orthogonal vectors prove that (a)…

A: Given r = i cost cosht + j sint sinht

Q: 4. Let f(x, y, z) = e² + zy² and let C be a straight line segment connecting the points (4,2,1) and…

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Q: Find the work done by the force F (x; y) = xyi-(y-x)j over the straight line from (1,1) to (2,3)

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Q: d) Find for w = xy + yz + xz , x = u + v , y = u – v , z = uv dv %3D at the point (u, v) = (-2,3).…

A: Given, The  functions  W=xy + yz + xzx=u+v,y=u-v,z=uv and the point is (u, v) = (-2, 3).

Q: Find the work done by the force F= xyi + (y - x)j over the straight line from (-1,- 2) to (2,1).

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Q: 1. If F = (2z+ y,z, 2y+ x) and n is a unit vector in the y-direction, (V×F)·n equals (a) (0,1,0).…

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Q: 4. Find the work done by the force F(r,y,2) = (x,-z,2y) in moving straight from (0,0,0) to (1.1.0),…

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Q: 2. Find the directional drevative of f(x, y,z) xsinyz at the point (1,3,0) in the direction of i =i+…

A: Use projection formula

Q: Consider p(x, y, z) = 2xyz - x²y³ v(x, y, z) = xy² + y sin(z) F(x, y, z)= F₂(x, y, z)i + F₂(x, y,…

A: Let F→=f1, f2, f3. The gradient of the scalar field: g is defined as: ∇g=∂g∂x i^+∂g∂yj^+∂g∂zk^. The…

Q: Consider Y; = 0 + 0,X¡ + E¿, i = 1,2,..,n, e, - N(0,0), the cov(ô,Y) = %3D

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Q: F(x, y, z) = xyzi+ (2ry+2) j+y°z°k conservative?

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Q: F(1, y) = e'i+ (re + y)j. Show that F(I, y) is conservative. Find f(r. u) such that F(I. V) = Vf(r,…

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Q: 7. If r = (x, y, z) and F vector r) is there a value p so that div F = 0? HE (where r| denotes the…

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Q: If f(x, y, z) = x² y + y² z + z2 x for all (x, y, z) e R3 ar i+ j+ k, then the value %3D and V dy of…

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Q: The gradient vector of g(x, y) = at (√√2,-1) is Vg(√2, -1) is equal to Select one: O True O False -

A: Given: The gradient vector of g(x,y)=x22-y22 at (2,1) is ∇g(2,1) is equal to &lt;2,1&gt;