Module 2 Equations and Graphs
Graph:
The three dimensional graph, with three axes, labeled x, y, and z, shows a
curved line running through it, which represents the trajectory of a particle.
There is a vector labeled capital X as a function of time, running from the
origin (0,0,0), to a point on the curved line. The point that the vector ends at
is signified by a big red dot, which represents the particle at a given time in
history.
Modification to graph:
A vector labeled V as a function of time, lowercase t, which represents the
velocity of the particle, extends from the red dot, the particle, outwards on a
path that is tangent to the curved trajectory of that particle.
Equation:
Capital V as a function of time, which represents particle velocity, equals the
derivative of capital X as a function of time, which represents the vector
from the origin to the particle along its trajectory.
Equation:
X superscript n, representing the position of the nth particle, equals capital X
superscript lowercase n in parenthesis, all as a function of time, lowercase t.
Also, capital V superscript lowercase n in parenthesis, the velocity of the nth
particle, equals capital V superscript lowercase n, all as a function of time,
lowercase t. Lowercase n is an integer.
Equation:
Capital X equals capital X as a function of time, lowercase t, and lowercase
n, an integer. Capital V equals capital V as a function of time, lowercase t,
and lowercase n, and integer.
Equation:
Capital X equals capital X as a function of time, lowercase t, lowercase a,
lowercase b, and lowercase c, where lowercase a, b, and c must span the
space.
Equation:
The material coordinates lowercase a, lowercase b, lowercase c equal capital
X subscript zero, which equals capital X as a function of 0 and capital X
subscript 0.
Graph:
This three dimensional graph has three axes, the x, y, and z. It has a curved
line running through it, depicting the trajectory of a particle. There is a
vector traveling from the origin to the particle’s current position on it’s
trajectory. This vector is labeled capital bold-faced X as a function of time,
lowercase t. There is another vector traveling from the origin of the graph to
the beginning of the trajectory, at time, lowercase t, equal to zero. This
vector is labeled capital bold-faced capital X subscript zero.
Diagram:
This picture depicts a narrow tube of some kind, with a measuring device
stuck into it that resembles a right-angle ruler. An arrow is used to show that
fluid is passing through the tube in the direction of the arrow, past the
measurement device.
Diagram:
This diagram shows a device that bends the fluid as the fluid moves past,
while at the same time measuring the fluid particles. A dotted box surrounds
the fluid that is in the scope of the device’s measurements.
Variable:
The partial derivative of capital theta with respect to time, subscript
lowercase x subscript zero.
Equation: (All derivatives are done with respect to time, lowercase t, unless
otherwise noted)
The partial derivative of capital theta subscript lowercase x equals the partial
derivative of lowercase theta subscript lowercase x times the partial
derivative of time subscript lowercase x subscript zero plus the partial
derivative of lowercase theta with respect to lowercase x subscript lowercase
i, with the whole derivative having a subscript lowercase t, times the partial
derivative of x with a subscript i, with a subscript for the whole partial
derivative of x subscript zero.
Equation:
The partial derivative of time with respect to time subscript one. The partial
derivative of lowercase x subscript lowercase i with respect to lowercase t
with a subscript of lowercase x subscript zero equals partial derivative of
uppercase X subscript lowercase i with respect to lowercase t with a
subscript of lowercase x subscript zero, equals capital V subscript lowercase
i as a function of capital X subscript zero and time.
Equation modified (All derivatives are with respect to time, unless otherwise
stated):
The partial derivative of capital theta subscript lowercase x subscript zero
equals the partial derivative of lowercase theta subscript lowercase x, plus
lowercase v subscript lowercase i as a function of bold lowercase x and time,
lowercase t, times the partial derivative of lowercase theta with respect to
lowercase x subscript lowercase i, with the whole derivative having a
subscript of lowercase t.
Equation modification (All derivatives are with respect to time, unless
otherwise stated):
The partial derivative of capital theta equals the partial derivative of
lowercase theta plus lowercase v subscript lowercase i times the partial
derivative of lowercase theta with respect to lowercase x subscript lowercase
i.
The partial derivative of capital theta equals the partial derivative of
lowercase theta plus lowercase bold v dot the gradient operating on theta.
Equation modification:
The partial derivative of capital theta in the previous equation is replaced
with capital D lowercase theta with respect to lowercase t.