JNTU
JNTU B.Tech II MATHEMATICS - II Previous Question Paper (March - 2022 - R13)
Answer any five questions. All questions carry equal marks.
Main
Show that the vector field given by A = 3x2yi + (x3 - 2yz2)j + (3z2 - 2y2z)k is irrotational but not solenoidal. Also find its scalar potential φ(x, y, z).
Answer
A vector field A is irrotational if ∇ × A = 0. Calculating the curl:
Calculating the partial derivatives: ∇ × A = i(-4yz - (-4yz)) - j(0 - 0) + k(3x2 - 3x2) = 0i + 0j + 0k. Since the curl is zero, the field is irrotational.
A vector field is solenoidal if ∇ · A = 0. Calculating the divergence:
Since ∇ · A ≠ 0, the field is not solenoidal.
To find the scalar potential φ, we solve ∂φ/∂x = 3x2y, ∂φ/∂y = x3 - 2yz2, and ∂φ/∂z = 3z2 - 2y2z. Integrating ∂φ/∂x gives φ = x3y + f(y, z). Differentiating with respect to y: ∂φ/∂y = x3 + ∂f/∂y = x3 - 2yz2, so ∂f/∂y = -2yz2. Integrating with respect to y: f(y, z) = -y2z2 + g(z). Thus φ = x3y - y2z2 + g(z). Differentiating with respect to z: ∂φ/∂z = -2y2z + g'(z) = 3z2 - 2y2z, so g'(z) = 3z2, which gives g(z) = z3 + C. The scalar potential is φ(x, y, z) = x3y - y2z2 + z3 + C.
Evaluate the integral of F · n dS where F = 6zi - 4j + yk and S is the portion of the plane 2x + 3y + 6z = 12 in the first octant.
Answer
Using the surface integral formula for the plane 2x + 3y + 6z = 12, we have z = (12 - 2x - 3y)/6 = 2 - x/3 - y/2. The normal vector n dS = (-(∂z/∂x)i - (∂z/∂y)j + k) dx dy = (1/3 i + 1/2 j + k) dx dy. The vector field F = 6zi - 4j + yk becomes F = 6(2 - x/3 - y/2)i - 4j + yk = (12 - 2x - 3y)i - 4j + yk.
The dot product F · n dS = [1/3(12 - 2x - 3y) + 1/2(-4) + 1(y)] dx dy = (4 - 2x/3 - y - 2 + y) dx dy = (2 - 2x/3) dx dy. Integrating over the region R in the xy-plane bounded by x=0, y=0, and 2x + 3y = 12 (or y = 4 - 2x/3):
Evaluating the integral: [8x - 2x2 + 4x3/27]06 = 48 - 72 + 864/27 = -24 + 32 = 8.
Using Green’s theorem evaluate ∫ (yi + xj) · dr, where C is a closed curve in the xy-plane enclosing an area A.
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State the Stoke’s theorem and divergence theorem. If F is irrotational, then find ∫ F · n dS, where S is an open surface.
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Find Fourier sine transform of f(x) = 1 / (x(x2+a2)) and hence deduce cosine transform of 1 / (x2+a2).
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Find the fourier series of f(x) = ex in (-π, π).
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Find the Fourier series expansion of the function f(x) = 0 for -π ≤ x ≤ 0 and f(x) = sin x for 0 ≤ x ≤ π and deduce 1/(1·3) - 1/(3·5) + 1/(5·7) + ... = (π-2)/4.
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State and prove change of scale property of Fourier transform.
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Find sin 52° using Newton’s interpolation formula from the following data: x: 45, 50, 55, 60; sin x: 0.7071, 0.7660, 0.8192, 0.8660.
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Write the normal equations to fit a curve of the form y = abx from the given data points (xi, yi), i=1, 2, ...n.
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Fit a curve y = a + bx + cx2 to the following data: x: 1, 3, 5, 7, 9; y: 2, 7, 10, 11, 9.
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Fit a straight line to the following data giving weights to x as 1, 1, 2, 1, 1 by the method of least square: x: 0, 1, 2, 3, 4; y: 1, 1.8, 3.3, 4.5, 6.5.
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Find the inverse of the matrix [[2, 3, 1], [1, 2, 3], [3, 2, 1]] using LU decomposition method.
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Find the root of 2x - log10x = 7, correct to five places of decimal using iteration method.
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Using the shooting method, solve the first boundary value problem u'' = u + 1, 0 < x < 1; u(0) = 0; u(1) = e - 1.
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