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Introduction to Ocean Dynamics
Warm-Ups (Please return your answers on the 2nd class: Sep/28/2012)
Review of Vector Algebra, Calculus and Differential Equations
1. The "cross product (×)" of 2 vectors A and C is also a vector which I shall call "B", i.e. B = A × C.
If vector A points eastward (in positive x-direction) and vector C points upward (in positive z-direction), which
direction does the vector B = A × C point?
(a) westward;
(e) no direction;
(b) northward;
(c) southward;
(d) downward;
(f) ____________________ (YOA - your own answer).
2. What is the magnitude of vector B, |B|, in term of |A|, |C| and angle between A and C = θ?
(a) |B|=|A|tan(θ)/|C|;
(e) |B|=|A|-|C|;
(b) |B|=|A||C||sin(θ)|; (c) |B|=|C||A|;
(f) ____________________ (YOA).
(d) |B|=|A|+|C|;
3. The "dot product ()" (also called "scalar product") of 2 vectors A and C is a scalar which I shall call "b", i.e. b =
AC. It is given by:
(a) b = |A||C|;
(e) b = |A||C| cosθ;
(b) b = |A||C|θ;
(c) b = (|A|+|C|) sinθ;
(f) ____________________ (YOA).
(d) b = |A||C| coshθ;
4. Let k be a unit vector (i.e. its magnitude |k|=1) pointing upward (z-direction), and let u be any horizontal vector,
then what is k × (k × u) equal to?
(a) 0;
(e) k|u|;
(b) |k||k|u;
(c) k × u;
(f) ____________________ (YOA).
(d) -u;
5. Let a and b be 2 vectors in 3-dimensional Cartesian space (x1, x2, x3): a = i1a1 + i2a2 + i3a3 and b = i1b1 + i2b2 + i3b3,
where i1 is the unit vector in x1-direction, i2 is the unit vector in x2-direction, and i3 is the unit vector in x3-direction.
Then what is ab equal to in term of a1, a2, a3, b1, b2 and b3?
(a) ab = a1a2a3b1b2b3;
(b) ab = a12+b12+a2b2+a3b3;
(c) ab = a1b1+a2b2+a3b3;
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(d) ab = a1b2+a2b1+a3b3;
(e) ____________________ (YOA).
[Note that we often also write (x1, x2, x3) as (x, y, z) and (i1, i2, i3) = (i, j, k), but writing with subscripts "1", "2" and
"3" is sometimes very convenient].
6. With the unit vector (i, j, k) defined as in above problem, write down the answers for the following:
i  i = _____;
j  k = _____;
k  k = _____;
i  j = _____;
j  j = _____;
k  i = _____;
i × i = _____;
j × j = _____;
k × k = _____;
i × j = _____;
j × k = _____;
k × i = _____;
i × k= _____;
j × i = _____;
k × j = _____.
Note the following simple rule for the cross products in the last 2 lines of the above problem: the answer is
positive if the order of (i, j, k) is anticlockwise (as shown below), and is negative if the order is clockwise:
7. The gradient operator ∇ in three-dimensional (3D) space is defined as ∇ = i∂/∂x + j∂/∂y + k∂/dz. Similarly, in
two-dimensional (2D) xy-space, it is ∇H = i∂/∂x + j∂/∂y. Suppose we have a function "T" in 2D: T(x,y) - for example
"T" can be the temperature of the ocean's surface at a given time. Write down what "∇HT" is in terms of i∂/∂x,
j∂/∂y and T:
∇HT = ________________________________?
Is ∇HT a
scalar
or
vector
or
__________( YOA)?
(Choose one or YOA).
8. Which direction is ∇HT in the following choices: choose the correct one, A, B, C, D, E, or F?
9. In the followings, evaluate dF/dx or ∂F/∂x for the given function "F". For example, F = ye2x, ∂F/∂x = 2ye2x.
a. F = 2x3,
dF/dx = __________________________________.
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b. F = a.yx.x-3, a=constant,
∂F/∂x = __________________________________.
c. F = 4.e-2x + sin(x) + ln(x),
dF/dx = __________________________________.
d. F = b.sin(x).e-2x + cos(3xy), b=constant,
∂F/∂x = __________________________________.
e. F = sin(x2)/cos(x),
dF/dx = __________________________________.
f. F = yx,
∂F/∂x = __________________________________.
10. In the followings, evaluate the indefinite integration ∫F(x)dx for the given function "F". For example, F = ye2x,
∫F(x)dx = ye2x/2. (Strictly speaking, an integration constant should be included - but you can omit it for this
exercise).
a. F = 2x3,
∫F(x)dx = __________________________________.
b. F = sin(2x),
∫F(x)dx = __________________________________.
c. F = a/x,
∫F(x)dx = __________________________________.
d. F = a/x2,
∫F(x)dx = __________________________________.
e. F = exp(yx),
∫F(x)dx = __________________________________.
f. F = sin(x).cos(x),
∫F(x)dx = __________________________________.
11. Chain rule: for any two functions u(x) and v(x),
d(uv)/dx = __________________________________.
𝑏
𝑏
12. From (11), therefore, d(uv) = udv + vdu. Then, ∫𝑎 𝑢𝑑𝑣 = [𝑢𝑣]𝑏𝑎 − ∫𝑎 𝑣𝑑𝑢, which is sometimes called the
integration by parts. Apply this to show that:
∞
∫0 𝑒 −𝑥 cos(𝑥)𝑑𝑥 = 1/2.
13. Solve the following differential equations. Include any integration constants or functions.
a. ∂F/∂x = x2,
F(x) = __________________________________.
b. d2F/dx2 + ω2F = 0,
F(x) = __________________________________.
c. d2F/dx2 - ω2F = 0,
F(x) = __________________________________.
d. d2F/dx2 + ω2F = x,
F(x) = __________________________________.
14. By trial and error, guess what the general solution to the following partial differential equation is:
∂u/∂t + c.∂u/∂x = 0, for c = constant and u a function of (x, t).
15. Then also obtain the general solution of the following partial differential equation:
∂2u/∂t2 - c2 ∂2u/∂x2 = 0, for c = constant and u a function of (x, t).
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∞
16. Show that:
∫0 𝑒 −𝑥 sin(𝑥)𝑑𝑥 = 1/2.
17. Show that:
∫−∞ 𝑒 𝑥 sin(𝑥 + 𝜋/4)𝑑𝑥 = 0.
18. Show that:
∫−∞ 𝑒 𝑥 cos(𝑥 + 𝜋/4)𝑑𝑥 = 1/√2.
19. Solve:
d/dt + if = 0, with initial condition (t=0) = o
20. Solve:
d/dt + (if+r) = 0, with initial condition (t=0) = o
21. Solve:
d/dt + if = A.exp(-ift), with (t=0) = o, and “A” a constant.
0
0
22. Solve for X & Z: dX/dt = cos(kx-ωt);
dZ/dt = sin(kx-ωt),
23. What is:
cosh(x)  ?
as x ~ ∞
24. What is:
sinh(x)  ?
as x ~ ∞
25. What is:
cosh(x)/sinh(x)  ?
as x ~ ∞
26. What is:
sinh(x)/sinh(x)  ?
as x ~ ∞
27. What is:
cosh(x)  ?
as x ~ 0
28. What is:
sinh(x)  ?
as x ~ 0
29. What is:
cosh(x)/sinh(x)  ?
as x ~ 0
30. What is:
sinh(x)/sinh(x)  ?
as x ~ 0
31. Show that:
∫0 cos(x) . sin(𝑥)𝑑𝑥 = 0.
32. Show that:
∫0 𝑠𝑖𝑛2 (𝑥)𝑑𝑥/(2𝜋) = 1/2.
33. Show that:
∫0 𝑐𝑜𝑠 2 (𝑥)𝑑𝑥/(2𝜋) = 1/2.
k &  are constants.
2𝜋
2𝜋
2𝜋
34. Show that the general solution to:
𝑑4 /𝑑𝑥 4 − 𝑑/𝑑𝑥 = 0
is:
 = C1 + C2 exp(m2 x) + C3 exp(m3 x) + C4 exp(m4 x)
where
m2 = 1, m3 = exp(i/3)=(1+i3)/2, m4 = exp(i2/3) =(1i3)/2
and
the C’s are arbitrary constants.