Show that ∇2(rn) = n(n+1)rn-2

Given r = √(x2 + y2 + z2), we have r2 = x2 + y2 + z2. Differentiating partially with respect to x...

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JNTU B.Tech II MATHEMATICS-II Previous Question Paper (August/September - 2022 - R13)

B.Tech | II | II | R13 | August/September - 2022

Answer PaperPaper Code: 114DDMaximum Marks: 75Time: 3 hours

Answer any five questions. All questions carry equal marks.

Main

115 marks

Show that

∇²(rⁿ) = n(n+1)r^n-2

7 marks

Answer

Given r = √(x² + y² + z²), we have r² = x² + y² + z². Differentiating partially with respect to x, y, and z gives ∂r/∂x = x/r, ∂r/∂y = y/r, and ∂r/∂z = z/r.

First, calculate the gradient ∇(rⁿ):

∇(rⁿ) = n r^n-1 (∂r/∂x i + ∂r/∂y j + ∂r/∂z k) = n r^n-1 (x/r i + y/r j + z/r k) = n r^n-2 r̅

Next, find the Laplacian ∇²(rⁿ) = div(∇(rⁿ)):

∇²(rⁿ) = ∇ · (n r^n-2 r̅) = n [ (∇r^n-2) · r̅ + r^n-2 (∇ · r̅) ]

Since ∇r^n-2 = (n-2)r^n-4 r̅ and ∇ · r̅ = 3:

∇²(rⁿ) = n [ (n-2)r^n-4 (r̅ · r̅) + 3r^n-2 ] = n [ (n-2)r^n-4 r² + 3r^n-2 ] = n [ (n-2)r^n-2 + 3r^n-2 ] = n(n+1)r^n-2

Evaluate

∫∫_S F.n ds

where

F = (x+y²)i - 2xj + 2yz k

and S is the surface of the plane

2x + y + 2z = 6

in the first octant.

8 marks

Answer

To evaluate the surface integral using the Divergence Theorem, we define the volume V bounded by the plane 2x + y + 2z = 6 and the coordinate planes in the first octant. The Divergence Theorem states:

∫∫_S F · n ds = ∫∫∫_V (∇ · F) dV

Calculate the divergence of F = (x + y²)i - 2xj + 2yzk:

∇ · F = ∂/∂x(x + y²) + ∂/∂y(-2x) + ∂/∂z(2yz) = 1 + 0 + 2y = 1 + 2y

The limits for the volume V are 0 ≤ x ≤ (6 - y - 2z)/2, 0 ≤ y ≤ 6 - 2z, and 0 ≤ z ≤ 3:

∫₀³ ∫₀^6-2z ∫₀^(6-y-2z)/2 (1 + 2y) dx dy dz

Integrating with respect to x:

∫₀³ ∫₀^6-2z (1 + 2y) [(6 - y - 2z)/2] dy dz = 1/2 ∫₀³ ∫₀^6-2z (6 + 12y - y - 2y² - 2z - 4yz) dy dz

Simplifying and integrating with respect to y and z yields the final result of 27.

Verify Green’s theorem for

∫_c (xy + y²) dx + x² dy

where c is bounded by

y = x²

and

y = x

15 marks

Answer

315 marks

Find the Fourier series of the periodic function defined as

f(x) = -x for -π < x < 0; x for 0 < x < π

and

f(x+2π) = f(x)

8 marks

Answer

Find f(x) if its sine transform is

e^-as

7 marks

Answer

415 marks

Find y(3.4) from the following table using Newton’s forward interpolation formula:

x	3	4	5	6
y	31	69	131	223

8 marks

Answer

Find y(6) if y(1) = 4, y(2)=5, y(7) = 5 and y(8) = 4 using Lagrange’s formula.

7 marks

Answer

Fit a parabola

y = a + bx + cx²

to the following data:

x	1	2	3	4	5	6	7
y	2.3	5.2	9.7	16.5	29.4	35.5	54.4

15 marks

Answer

615 marks

Interpret geometrically Newton Raphson method.

7 marks

Answer

Find a root of the equation

f(x) = log x + x - 2

using Newton Raphson method.

8 marks

Answer

Find y(0.1) and y(0.2) using Runge Kutta fourth order formula given that

dy/dx = x² + y

and y(0) = 1.

15 marks

Answer

Given that

y' = x² + y²

, y(0) = 1, determine y(0.1) and y(0.2) by modified Euler’s method.

15 marks

JNTU B.Tech II MATHEMATICS-II Previous Question Paper (August/September - 2022 - R13)

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