Transcript Fear of Fish: The Contaminant Controversy

Math & Physics
Review
MAR 555 – Intro PO
Created by Annie Sawabini
Topics

Coordinate Systems
 Vectors



Notation
Dot & Cross Products
Derivatives




Review
Partials
Del Operator
Gradient, Divergence, Curl
Motion – laws and equations
 Miscellaneous

Coordinate System
 Right
hand coordinate system
z [Up]
Position
y [North]
x [East]
w [upward]
Velocity
v [northward current]
u [eastward current]
Note: Ocean currents are named for the direction they are traveling (e.g. a
northerly current is moving water along the positive y-axis). This is opposite the
convention used for wind. A north wind blows FROM the north, along the
negative y-axis.
Vector Notation
a
 Scalars


Magnitude only
ex. Temperature or Pressure
 Vectors


a
a
a a
Magnitude and Direction
ex. Displacement = distance (scalar) plus
direction
c
a
b
c
b
a
a
+
b
=
b
+
a
=
c
Vector into Scalar Components
 Resolving
Vectors into Scalar
Components on a 2D coordinate system
sin = opposite
y
hypotenuse
a
Ø
ax
ax = a cos ø
ay
cos = adjacent
hypotenuse
x
tan = opposite
adjacent
ay = a sin ø
Vector Operations
 The

dot product (a.k.a. the scalar product)
Two vectors dotted together produce a scalar
a • b = a b cos ø
 The

cross project (aka. the vector product)
Two vectors produce a vector that is
orthogonal to both initial vectors
|a x b| = a b sin ø
b
a
Derivatives
 Derivative
= the instantaneous rate of
change of a function
dy
the change in y
dx
with respect to x
where y = f(x)
 Also written as
f´(x)
Derivatives
 Example:
 Remember
why?
Derivatives

Power rule:



Chain rule:



f(x) = h(g(x)), then
f´(x) = h'(g(x))* g'(x)
Product rule:


f(x) = xa, for some real number a;
f´(x) = axa−1
(fg)´ = f´g + fg´ for all functions f and g
Constant rule:


The derivative of any constant c is zero
For c*f(x), c* f´(x) is the derivative
Partial Derivatives
derivative – a derivative taken with
respect to one of the variables in a
function while the others variables are held
constant
 Partial
 Written:
f
x
fx
Partial Derivatives
 Example:

Volume of a cone:
• r = radius
• h = height

Partial with respect to r:

Partial with respect to h:
, The Del Operator
 The


Del operator
Written:
Note: i, j, and k are BOLD, indicating vectors. These
are referred to as unit vectors with a magnitude of 1 in
the x, y and z directions. Used as follows:
a = ax i + ay j + az k
Gradient

Gradient – represents the direction of fastest
increase of the scalar function


the gradient of a scalar is a vector
 applied to a scalar function f:
Divergence

Divergence - represents a vector field's
tendency to originate from or converge upon a
given point.


Remember: the dot product of two vectors (F and )
produces a scalar
Where F = F1 i + F2 j + F3 k
Curl

Curl: represents a vector field's tendency to
rotate about a point


Remember: the cross product of two vectors (F and
) produces a vector
For F = [Fx, Fy, Fz]:
Newton’s Laws of Motion

First Law


Second law


In absence of external forces a body will move at
constant velocity or stay at rest (or either depending
on reference frame)
Observed from an inertial reference frame, the net
force on a particle is equal to the rate of change of its
momentum F = d(mv)/dt. Or more simply force
equals mass times acceleration.
Third law

To every action there is an equal but opposite
reaction
Equations of Motion

Speed


Velocity = distance / time


rate of motion (scalar)
speed plus a direction (vector)
Acceleration

the rate of change of velocity over time
a = dv / dt

average acceleration
a = (vf – vi) / t

Force

mass * acceleration
F = m*a
Free body diagrams
 Use
to define all the forces acting on a
body
 Don’t forget to define your axes