Reference no: EM132247684
Questions -
1) If the vector F = (ax+3y+4z)i + (x-2y+3z)j + (3x+2y-z)k is solenoidal, then determine the constant a.
2) Show that the vector A is irrotational if the vector A = (x2-yz)i + (y2-zx)j + (z2-xy)k.
3) Find divergence F, where F = xy2i + 2x2yj - 3yz2k at (1, -1, 1).
4) Find the work done by the variable force vector F = 2yi+xyj on a particle when it is displaced from the origin to the point vector r = 4i + 2j along the parabola y2 = x.
5) Evaluate ∫F.dr from the point A(0, 0, 0) to B(1, 1, 1) at along the curve vector r = it + jt2 + kt3 given vector F = xyi - z2j + xyzk.
6) Find the value of grad ∅ at (2, -2, -1) for ∅ =2xz4 - x2y.
7) Find the Fourier series expansion of f(x) = x2 in the interval -π ≤ x ≤ π and hence deduce that 1/12 - 1/22 + 1/32 - 1/42 + ........................... = π2/12.
8) Find the Fourier series of the periodic function defined by in the interval f(x) = 2x - x2 in the interval 0 < x < 3.
9) Find the value of grad φ at (2, -2, -1) for 2xz4 - x2y.
10) Evaluate ∫CF→.dr, where F→ = xyi + (x2 + y2)j and C is the curve y = x2 - 4 from (2, 0) to (4, 12) in the XY plane.
11) If the vector F = (ax+3y+4z)i + (x-2y+3z)j + (3x+2y-z)k is solenoidal, then determine the constant a.
12) Find the Fourier series for f(x) = x2 in the interval and deduce that -π < x < π and deduce that n=1Σ∞(-1)n+1/n2 = π2/12.
13) Obtain the half-range sine-series of f(x) = x2 in the interval (0, π).
14) Find the constant term and coefficients of the first and second cosine and sine terms in the expansion of y from the given table
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x
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0
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1
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2
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3
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4
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5
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6
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y
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9
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18
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24
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28
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26
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20
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9
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15) Find the Fourier series expansion of f(x) = xin the interval -π ≤ x ≤ π and hence deduce that π/4 = n=1Σ∞(-1)n+1/(2n-1).