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    Self-Exercises 2

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    Self-Exercises 2

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    braydenhassan16
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    Self-Exercises 2 (1)

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    Self-Exercises 2

    Uploaded by

    braydenhassan16

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    © All Rights Reserved
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    of 1

    School of Mathematics & Applied Statistics

    MATH283 Advanced Engineering Mathematics and Statistics


    Mathematics - Self-help Exercises 2
    Functions of Several Variables
    ∂z ∂z
    1. Find and when
    ∂x ∂y

    (a) z = 3x2 + y 2 (b) z = 2xy − 3y + x2 (c) z = x2 cos xy


    x−y y
    (d) z = (e) z = sin(x + 2y) (f) z = arctan
    x+y x
    xy
    (g) z= (h) z = exy sin x (i) z = xy
    x+y

    ∂2z ∂2z
    2. Find , and verify that
    ∂x2 ∂y 2
    ∂2z ∂2z
    =
    ∂x∂y ∂y∂x
    when z = ex sin y .

    3. Show that the function


    z = ey (y cos x − x sin x)
    satisfies Laplace’s equation
    ∂2z ∂2z
    2
    + 2 =0
    ∂x ∂y

    4. The voltage transient E in a transmission line satisfies


    ∂E ∂2E
    =k 2.
    ∂t ∂x
    2
    Show that the function E = ae−ω kt
    cos ωx is a possible solution. Here a, ω and k are constants.

    5. Show that u = (x − 2t) sin(x − 2t) satisfies the wave equation


    ∂2u ∂2u
    4 = .
    ∂x2 ∂t2

    6. Given that u and v are functions of x and y defined by u = 2xy and v = x2 + y 2 , and that
    ∂f
    f (u, v) = uv 2 , express in terms of x , y , u and v . [ 2v 2 x + 4uvy ]
    ∂y

    7. The moment of inertia I of a cylinder of mass M , radius r and length h about a line through
    its center perpendicular to its axis is given by
    1
    I= M (3r2 + h2 ).
    12
    Determine the approximate percentage increase in the moment of inertia I if r and h are
    increased by 1% without any change in mass. [ 2% increase]

    8. The volume of a frustum of a cone is given by V = 13 πh(a2 + ab + b2 ) , where a and b are the
    radii of its ends and h is its height. If the radius of each end is increased by 4% and the height
    is decreased by 1% , determine the approximate percentage change in the volume.
    [ 7% increase]

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