Q.6 Show that: B(x-1, y+1) = B(x, y) y

Q: 2+i (a) Evaluate dz , along the line x=2y.

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Q: 2. Let j = Find the point on the line y x that is closest to j. %3D %3D 26

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Q: 2. Evaluate f. (y²dx + x?dy) where C is the triangle with vertices (1, 0), (1, 1), (0, 0).

A:

Q: Evaluate ſſs 3y dS where S is the portion of y^2 + z^2 = 4 between x = 3 and x = 3 - z. %3D

A: The surface is ∬S 3y dS=∬S1 3y dS+∬S2 3y dS + ∬S3 3y dS

Q: 3. If V =(3x'y-z,xz' + y",-2r?:'). Find grad div at the points (2,-1,0).

A: Given , V=< 3x2y-z, xz3+y4, -2x3z2 > To Find , grad divV  Let  V=< V1 ,V2 ,V3 > On…

Q: 2.Evaluate . xy+ds where C is the right half of x2 + y? = 16

A: Integrate with use parametric form

Q: Find Duf at P. 2 + 2. 5,3 4 f(x, y, z) = 6x³y°z*; P(1, – 2, 1); u = Duf i

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Q: // | *dædydzB = {(x, y, 2) =€ R® 0 <« < 1,0 < y < 1,0 < z < x} {(x, y, 2) =€ R° 0<æ < 1,0 < y < 1,0…

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Q: o Find the Vf at the given point 1.f (x,y,2)=x² + y² -2:2 +z In x,P(1,1,1)

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Q: f(x,y) = x² + 3xy +y² fonksiyonu için fex (2,3)]. [fyy (2,3)] – Fzy (2,3)]? =

A: The answer is, (-5) The detailed solution is as follows below :

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: d the values of az/ax and öz/ay at the point z - xy + yz + y' - 2 = 0, (1, 1, 1)

A: Given that  z^3-xy+yz+y^3-2=0 To find the partial derivatives at point (1,1,1)

Q: and v= x2y% if z- fcx,y) where show that dz dv u=X-y 出。是、会 dy 24

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Q: In Exercise, verify that S and T are inverses.

A: GivenS:R2→R2 by Sxy=4x+y3x+yT:R2→R2 by Txy=x-y-3x+4yTo check S  and T are inverse of each other.

Q: 4. (a) If y = A + Bx + Cx2 and yo, yı, y2 are the values of y corresponding to x = a, a + h, a + 2h…

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Q: 4) Suppose that T : R³ → R² is defined by T((x, y, z)) = (1+2x – z , 3 – x + 6y) Is T linear?…

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Q: 2. W =I+xy- Z finds Wzz WZX NJ 3.

A: Given W = 1 + xy2 - z2

Q: 3. Show that S4 has the presentation (x, y|x* = y² = (xy) = 1). %3D

A: To show: S4 has the presentation x,y| x4=y2=(xy)3=1

Q: 3. (z, y)(1.5x, 1.5y) A' will be located at ( B' will be located at ( C' will be located at (

A: We have given the scalar factor i.e . (x,y)=(1.5x,1.5y)

Q: (1,1) S[3x*+5y+i(x°-y°)] dz along y=x Evaluate (0,0)

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Q: b) = f ) then prove that x ux + y uy + z uz = 0 u

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Q: a) If W =(x,y), where x v²-g' and y 2vg calculate W+ %3D

A: We’ll answer the first part of this question since due to complexity. Please submit the question…

Q: Find Vq if (a) =In (r) %3D (b) p= In (-) %3D We know r= /x² + y² + z2

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Q: x2 B) If w = x".f then show that: %3D x. Wx + y. Wy = nw.

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Q: 2. Evaluateſff, y dV, where: E = {(x,y,z)|0 <x< 1,0 < y < 4x,y < z < x² + y} %3D

A: The solution are next step

Q: (2 fi = f(x²+xyz, y² +22= Prove that (y²-2x) ou 2ZX) then ne (²h-x) he 18

A: Concept: Partial derivative : The partial derivative is similar to the normal derivative except in…

Q: 1. If x, y, z E R, prove that (c) ||x| – |y|| < |x –yl. (d) \xy| < }(x² + y²)

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Q: ( b) For n=1,2,K , show that (x² –1)y"*) +2(nx + p)y") +(n² –n)y"-1) =0 where %3D d" y p") = (Hint:…

A: * Given that y = x+1x-1pa dydx = px+1x-1p-1ddxx+1x-1    dydx = px+1x-1p-1x-11-x+1.1x-12   dydx =…

Q: 2+i (a) Evaluate [z dz , along the line x=2y. 1

A: Since you mention to solve only part a therefore I will solve only (a)

Q: Gy Let Px) = x²/3, x. thens Rx) = x2/3 ×シ。 then: %3D O Find ê cx) ? 2 Gruphs off R and f

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Q: (i) If y =g (x1- x2). Let u = x1- x2, show that (a) ay/du = ây/Əxı=- dy/ax2 %3D (b) a@y/ðu² = y/Əx¡²…

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Q: T (x, y) (2x,y) is an is %3D

A: T(x,y)=(2x,12y) The transformation of any figure changes the size, shape and the position to create…

Q: Q: If u =f(x² +2yz, y² + 2=x)Prove that (y-ax

A: Step by step compute the derivative with respect to each variable x,y or z. Then substitute it in…

Q: 5Is there a value of r sotha X = 1 y = 2 and Z=r is asolution to the linear systen ? If there is ,…

A: Putting equations as;23-11-1241-2 xyz=11-712Augmented…

Q: (x + 2y), 0sx<1,0<y<1 fx: 0, elsewhere PG<x<를 &y<b a.

A: Given,f(x)=23(x+2y); 0≤x≤1 , 0≤y≤10; elsewhere

Q: If f (x, y) = V3 + y³ In (z +y°)e*in(zyty") +2² + 10y , then fz (1,0) = |3D %3D O 2 4. O 3

A: Given that fx,y=x3+y33lnx2+y2esinxy+y2+x2+10y3 First we have to find fxx,y where fx=∂f∂x

Q: 2. 2sin(хy) at x D 2, у3D Зя/4 3. z= In(x + 1) + y° at the point (0,3,9)

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Q: 9(x, y, z) = xy + xz + yz; C: r(t) = (1, 21, 3t) , for 0<i54 %3D

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Q: PART C dy 12. Show that dx 2 for the relation %3D V%3D10, x, y X. # 0. y

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Q: if y³ - y = 2x Then, d'y -24y prove that dx (3y -1) |

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Q: Exl Let c R,Z) be usual toP s Pace and A-fcxy)ER/ x²+y²< 13 find A Ext CA), Ā , A°

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Q: 3. If (x, y, z) = xy²z find grad 6 at (2,−1,3).

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Q: Express x = In(3t), y = 5 – t in the form y = f(x) by eliminating the parameter. %3D y =

A: Given that x=ln(3t) and y= 5-t we need to write y=f(x) by eliminating the parameter  Now , x= ln(3t)…

Q: | 2.xdV where E = {(x, y, z) |0 < y < 2, 0sxs 4- y²,0<z< y}

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Q: Q,3 Evaluate [II, xyz dxdydz, 3 < x S-2,0 < y S 1,0 S zs-

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Q: 16. f(x, y, 2) = y - Vr? + z?; P(-3, 1, 4); a 2i-2j-k

A: To find the directional derivative of f at P in teh direction of a So, given f(x,y,z)=y-√(x²+z²);…

Q: Find Duf at P. 2. f(x, y, z) = 6x°y°z*; P(1, – 2, 1); u = i+ 1 Duf = i

A: We need to find directional derivative at p.

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.