Evaluate F.dr, where F(x,y)=[2y-3cos(x)]i + [4x +6sin(y)]j and C goes counterclockwise around the triangle with vertices (0,0). (1,0), and (0,1).

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

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Q: Find Vf at the given point. f(x, y, z) = 32³ – 2(x² + y²) z + tan '(4xz), (0, 2, 3) Vf\(0,2,3) f(x,…

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Q: Find the linearization L(x, y) of the f(x, y) = x°y® at the point (2, 2). L(x, y) = %3D

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Q: (a) Let F = 4x27 + y} - (z – 1)k. Compute I7. dA where S is a square of side 5 units centered at…

A: Given that, the vector is F→=4x2i→+yj→-z-1k→. Determine the value of ∫SF→·dA→.…

Q: Find div F and curl F if F(x,y, z) = xzºi+ 5y°x²j+ 5z?yk.

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Q: Find the linearizations of the functionsat the given points. ƒ(x, y, z) = √2 cos x sin ( y + z) at…

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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: what is the area of the rotational surface formed by rotating the part of the function y= x^3/3…

A: We apply the above formula to solve this

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

A: Given u = x2-y2v = 2xy Then ux = 2xuy = -2yvx = 2yvy = 2x

Q: Find the linearization L(X,Y) of the function  f(x,y)=y-1/x+1 at(0,0).

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Q: 3. Verify that u(x, y) = x² + 4.x – y² + 2y is harmonic and find %3D - a) the harmonic conjugate…

A: See the attachment

Q: w In(x2 + y2 +z?), x = ue" sin u, y = ue" cos u and z ue", using chain rule at the point (u, v) =…

A: Given w=lnx2+y2+z2x=uevsinuy=uevcosuz=uev

Q: - Find div F and curl F if F(x, y, z) = 10y¹z7i – 17xªzºj – 3xyễk. div F= curl F=

A: The general form of a vector-valued function in three dimensions is given by F=F1i+F2j+F3k, where…

Q: Find a linearization L(x, y, z) of f(x, y, z) = x²y + 4z at (1,-1,2).

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Q: Study the analytically of w = L_r(z) at the points z = 0, z = v2,z = -2i

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Q: 6. Let f(r, y, z) = e=² cosh y + z ar ff at the point P in the direction of :

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Q: Find the linearization L(x,y,z) of the function f(x,y.z) at the given points. f(x.y.z) = e* + cos (y…

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Q: Let f(x, y, z) = ln(x² + z) + y²e®z – cos(2yz). Find fzyn. Zyx ·

A: Given function is  f(x, y, z)=ln(x2+z)+y2exz-cos(2yz)

Q: Determine Taylor's formula of order 1 and center a = (0, 0) of f(x)=ex1sin(x2).

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

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Q: dz Use chain rule to find at (u, v) = (1, 1) if dv z = z sin(y² – z + 1)4), e = 3u – v², y = vº.

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Q: 6. Let f(x,y) = sin(x + y) + arcsin (xy) a. Find the linearization L(x,y) at the point (0, ")

A: We have to find linearization of given function f(x,y): Concept: The linearization of a function…

Q: Using an appropriate linearization, the local linear approximation of f(x)%3D+ cos r at is 2.

A: To determine the linear approximation of the function.

Q: (a) Evaluate f. (sin x + 3y²)dx + (2x – e-y*)dy, where C is the boundary of the half-disk x² + y? <…

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Q: Find and ди ди using the chain rule. w(x, y) = 14x²y², x = e" cos v, (u, υ) ди (u, υ)

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Q: Solve by theorems of linearity and translation. f(t) = cos t + sin t

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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: find the linearization L(x, y) of the function ateach point. ƒ(x, y) = ex cos y at a. (0, 0), b. (0,…

A: Consider the following function: fx,y=excosy Differentiate fx,y with respect to x taking y as…

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

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Q: Using an appropriate linearization, the local linear approximation of f(x) n+ cos a at is %3D

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Q: 1. (a) Let f(r,y)- cos(z + 2y). Compute the diectional derivative Df(r,9), where u- (b) Compute ff,…

A: Solve the following

Q: If f(x, y, z) = ln(æª + y!0 + z®) + V(x10 + y8 + zª), then find fr, fy and f; at (1, 1, 1).

A: Given, f(x,y,z)=ln(x4+y10+z8)+(x10+y8+z4)

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 f(x, y) = xcosy- y'e*y. fxy at (1,0)is equal to: %3D -2/3 -2 1 0.

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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: =

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Q: 2- Show that the function U(x,y)= 3x²y-y² + y is Harmonic and then find it's Harmonic conjugate V(x,…

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Q: Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y)…

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost other…

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: The linear approximation to f(x, y) = /2y + cos2(x) at the point (0,0) takes the form Ax + By + C.…

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Q: The conjugate harmonic function v(x, y) of u(x, y) = In (x2 + y2) is *() + 1. 2tan -1 2tan-1 + C ()…

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

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Q: 1. Use a change of variables to evaluate [[ y- dA where R is the trapezoid with vertices (1,0,…

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Q: Find the flux of F = (x - y)i + xj across the circle x2 + y2 = 1 in the xy-plane.

A: Given F = (x - y)i + xj  across circle x2+y2=1

Q: X+Z+3Siny X +2y-2

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Q: Find div F and curl F if F(x, y, z) = 16y°z³i – 9x°z°j – 7xy'k. - - div F= curl F=

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Q: 6x Find the linearization L(x,y) of the function f(x,y) = e cos (y) at the points (0,0) and

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Q: (a) Let F z sin y, where r(s,t) = st and y(s, t) = st. Use the chain rule to find %3D %3! OF

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Q: Show that the function f(x,y) =x² cos² y+ ysin² x+xe" , satisfies the 2 COS Lipschitz condition in…

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Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.