R and r are defined by E(; - 1)(y, – I) VE(*; - 1)² E(Y SSR Sry S,, Syy R? TY r = SST' (1) Show R2 = r2.

Q: az xdz + Y- = xy+z. a) If z = xy - xei , show that x- ду

A: Since you have posted multiple questions in this question which are not interlinked so I have solved…

Q: (b) f(x) = where A; E S™ – (det(Ao + x1A1 + ...+ xn An))/m, on {x | Ao + ¤1A1 ++xnAn > ► 0},

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Q: Iffis of class c? and z= f(y.x), con u= x°y°,v = 5x+1. Verify that| :4xy kxy+ 4x³y° + 10xy- af дхду…

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Q: 8.If V=k r°, and V=4 when r=2 if dr/dt =2, find dV/dt ind th

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Q: if x²+y²=t-- and x*+y*=t*+÷,prove dy dx x'y

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Q: Q7 Show that CX,y) =e is Condinuous at co,0)

A: The given function is:  fx,y=e-x2+y2 The given point is 0,0

Q: Find dY at x = -1ifY zu%5V=4 3. and u= X²+ X

A: Given that y=u3+5u-4 and u=x2+x We need to find dydx at x=-1 Take y=u3+5u-4 Substitute "u"…

Q: dy Find at t = a where dt 1 7. у %3D = 3s + 2 and s t2 when a =1. + 4

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Q: ds if s = In [t*(t² – 1)]| dt 20. Find

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Q: -2+z) ds over t e point (0, 0, 0)

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Q: If r + s2 + y3 = 12, dr/dt = 4, and ds/dt = -3, find dy/dt when r = 3 and s = 1

A: We are given the relation by

Q: Suppose that u and v are differentiable at x = 2 and that u(2) = 4, v(2) = -9, u'(2) = 5, and v'(2)…

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Q: If x2 + y? + z2 = 9, dt dy = 9, find dt dz when (x, y, z) = (2, 2, 1). = 4, and %3D dt %3D

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Q: if x²+y²=t-- and x*+y*=t*+=,prove dy X t dx 3. ху

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Q: dw If w = f(x,y,z) = In(x² +y² z?),x = cost y= sin t ,z = 4VE , then is : dt + ww.

A: Given  w = f(x,y,z) = ln(x2 + y2 + z2) x = cos(t) , y = sin(t) and z = 4t Substituting the values of…

Q: Verify Young's Theorem y? x2 z = In(2x + y) – + e(x+2y) + - y

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Q: 6. Ifw = f(x-y, y-x), then show that, ar + L = 0. dx ay

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Q: If the new DE is written in the form M(x, y) dx + N(x, y) dy = 0, one has My = || N, = X, Since M,…

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Q: e solution of the differen uation dy/dt 3 y with in ndition y(0) = 5 is %3D t

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Q: if x² + y² =t -- and x* + y* = t dy prove dx 1

A: Given, if  x2+y2=t-1t and  x4+y4=t2+1t2 To prove that  dydx=1x3y

Q: If y In(z- y) then y at the point where y = 0 is equal to

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Q: Suppose w = x = cos? t, y = sin? t, z = dw Evaluate at t = 3 , by using the Chain Rule dt

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Q: Find the value of ds/du at u = 2 if s = t^2 + 5t and t = (u2 + 2u)^(1/3).

A: Find ds/dt from s=t^2+5t

Q: If a and ß are positive constants, show that the particular solution of dy %3D У (В — ау) dt that…

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Q: If x"y" = (x+y)** m+n m. X' d'y prove that = 0 dx?

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Q: Consider h(x,y, z) = cos (xy) + eY² + In (xz) at the point P ( 1,0, = 1 Deter-

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Q: 1. Use the Chain Rule to find u and ou for w = x²y3 where x = dw at s2 +t2 and y = ds

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Q: 2. If f(x,y)= te Vy of=D0 then show) that 49 c) of af =0 dxay

A: Consider the given function. fx,y=1yex-y24y (i) First, we have to show that ∂2f∂x2-∂f∂y=0. First,…

Q: Eval S fCx,y,z)dV Por f auel regien W M, fory2)= 2x W: x'+ y? < zs 4

A: The region W : x2+y2≤z≤4, and f(x,y,z)=2x, we have to evaluate the value of ∬∫Wf(x,y,z)dV.

Q: dy (3) If y = | (t – 1)²dt for x > 0, then compute dz In z

A: fundamental calculus theorem 2 states that if f is continuous on [a, b], then  ∫abf(x)dx=F(b)-F(a)…

Q: 4. Let z = x2 + y³, x = 2st, y = s - t2, s = u", and t = u sin w. Compute for Oz ду and дw ди

A: Let z=x2+y3, x=2st, y=s-t2, s=uw, and t=u sin w.To compute for ∂z∂w and ∂y∂u.

Q: 27. Consider the piecewise path which goes along (t,3 – 3t, -t) for 0 <t<1, then along (1,t,-1+ t)…

A: Line Integral along the given parametric curve.

Q: If x2 + y2 = z2, find dy/dt when x = 3, y = 4, dx/dt = 2, and dz/dt = 2. (Enter your answers as a…

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Q: e) Suppose that W= √√x² + y² + z², x= cos 0, y = sin 0, z = tan 0. dw T Use the chain rule to find…

A: According to our company's guidelines I can only answer (e), since (e) and (f) are not inter…

Q: Suppose that the joint p.m.f. of Y and Y2 is defined by: 2x + y for x = 1, 2,3; y = 1,2,3 54 f(x, y)…

A: The probability mass function of Y1 and Y2 is,

Q: O from the equation z = In(x² + y²) + Zyy Show that zxx %3D

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Q: : 0s 0s n, Os zs 1, 6Vzs rs 6}. Let E= Compute y?e(x2+y?)³ dv•

A: Given that, Let E=r,θ,z : 0≤θ≤π, 0≤z≤1, 6z≤r≤6 compute ∬∫Ey2ex2+y22dV

Q: 8. Let f(x, y) = cos(y – x) and r(t) = (e', e2t). Find if (r(t)) at t = ln(3).

A: The solution is given as

Q: 2. Compute ¡ E x dV , where E “ tpx, y, zq | x ď 0, z ď 0, x2 ` y 2 ` z 2 ď 9u

A: The given integral,

Q: If W = xyz with x = 2t*, y = 3t-, and z = 4t¬³, what is dw dt 24 O In(t)

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

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Q: B. Let z- ( x-y)e×y (x-y)exy Show: Zxx + zyy + exy Cya-x3)= -z (xy+ 2)

A: Partial derivative of a function z=f(x,y) with respect to x is the derivative of z keeping y…

Q: 1. Let z = f(x, y), find for: and dy - - (a) e"y + z ln y = 0 (b) yz – sin z = xy

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Q: S, (x + 2y) dx + x dy. C consists of line segments from (0. 0) to (2. 1) and from (2, 1) to (3, 0)

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Q: 4] I Let w = 5 cos(xy) – sin(xz); x = 1/t, y = t, z = dw using an appropriate form of the chain…

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Q: If z = e* sin y, where x = s?t2 and y = s²t, by using Chain Rule find ôz/ðs and az/dt.

A: Let's find.

Q: d If g (x) =-=-^[e'- In(2t+1)]dt, find g''(x) . dx -2 A ex-In(2x+1) 4 B et + (2x + 1)2 4 (2x+1)2² 2…

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Q: Let z = f(x, y) and suppose x = 2u – 3v and y = 5u + 6v. Use the chain rule to express dz in terms…

A: x=2u-3vy=5u+6v Now we apply the chain rule  ∂z∂u=∂z∂x·∂x∂u+∂z∂y·∂y∂u Take derivative of x  and y…

Q: Evaluate 2.xyz dx + x²zdy +x²ydz over the path r(t) = (t², sin(nt/4), e'“-2') for 0 <t<2.

A: The given integral is: I=∫C 2xyz dx+x2z dy+x2y dzOver the path:  r(t)=t2,  sinπt4,  et2-2t    for…

Q: If a and ß are positive constants, show that the particular solution of dy %3D У (В — ау) dt that…

A: Given that dydt=yβ-αy and y=y0 at t=0 We have to find the solution of the above equation.