Class on Januuary 28 2019
Chris introduced students to useful vocabulary that would be relevant throughout the class:
For example, in this sample above, team North measured:
Temperature: 28.2 degrees C (scalar)
Salinity: 26 ppt (scalar)
Movement (flow): 11 cm/minute to the northeast (vector)
Volume: 75 cm3 (scalar) *
Location: 13 feet east, 9 feet south from northwest corner of the simulated bay (the classroom)
* volume calculation: average radius of the cup where water was present was 2.30 cm,
height of the water: 4.5 cm, therefore:
area of average horizontal slice = π*radius2 = 3.14159x2.3x2.3 = 16.61 cm
and volume = 16.61 x 4.5 cm = 75 cm3
Chris then accumulated the results on the blackboard and discussed the behavior of the overall simulated bay based on measured samples:
generally, the ocean water comes in via the east passage and flushes out via the west passage.
Chris ran a class exercise for students to gain more exposure to vectors. He described a unit vector as comprising of three components: e1 is length of 1 along x1, e2 is length of 1 along x2, e3 is length of 1 along x3.
He then set up a 25-foot string which cut at a 37-degree diagonal across the room, started at the floor, and ended at 5 feet height. He identified two ways that vectors are represented notationally to be unique:
1. The <u1, u2, u3> notation (known as u with a tilde '~' accent above) identifies three measurements along three axes, from the 0,0,0 origin, using x, y, and z as typical axes representation. The string could then be identified as a vector by <20, 15, 5> (20 feet on the x (east), 15 feet on the y (north), and 5 feet on the z (upward).
2. The vector could also be represented by its length and an angle in 3-D space (25 feet, 37 degrees to the NE)
which is known as ui notation
note: the 37 degree angle is made up of 48.6 degrees North from East axis, and 14.5 degrees Up from the East axis.
Chris asked students to participate in a speed game whereby each team had to identify a unique vector based on the <u1, u2, u3> notation. For example <15, 15, 0> was represented by:
Chris handed out a homework assignment for students to work on for Friday's upcoming class. To prepare them, he described the dot product between two vectors A and B:
A . B = A1 * B1 * e1.e1 + A2 * B2 * e2.e2 + A3 * B3 * e3.e3 = A * B * cos(angle between them)
(each e#.e# term evaluates to 1)
He said a good reminder of dot product is Jiminy Cricket while the cross product (A x B) is Honest John (to be elaborated upon next class session).
To warm students up to the cos() calculation, Chris mentioned that the cosine of 0 degrees is 1 and the cosine of 90 degrees is 0.
The homework assignment would be continued in class on Friday, including an introduction to using MATLAB in supporting calculation.
- Continuum mechanics — making small fluid parcels (compared to overall domain) to model a body of water such that: atoms < parcel size < scales of domain
- scalar — a measurement that has just a magnitude
- vector — a measurement that has magnitude and direction
- tensor — a mathematical object represented by an array of components that are functions of the coordinates of a space whereby:
- rank 0 tensor is a scalar
- rank 1 tensor is a vector
- rank 2 tensor is a stress tensor (magnitude and two directions)
- rank 3 tensor is Levi-Civita (magnitude and three directions)
For example, in this sample above, team North measured:
Temperature: 28.2 degrees C (scalar)
Salinity: 26 ppt (scalar)
Movement (flow): 11 cm/minute to the northeast (vector)
Volume: 75 cm3 (scalar) *
Location: 13 feet east, 9 feet south from northwest corner of the simulated bay (the classroom)
* volume calculation: average radius of the cup where water was present was 2.30 cm,
height of the water: 4.5 cm, therefore:
area of average horizontal slice = π*radius2 = 3.14159x2.3x2.3 = 16.61 cm
and volume = 16.61 x 4.5 cm = 75 cm3
Chris then accumulated the results on the blackboard and discussed the behavior of the overall simulated bay based on measured samples:
generally, the ocean water comes in via the east passage and flushes out via the west passage.
Chris ran a class exercise for students to gain more exposure to vectors. He described a unit vector as comprising of three components: e1 is length of 1 along x1, e2 is length of 1 along x2, e3 is length of 1 along x3.
He then set up a 25-foot string which cut at a 37-degree diagonal across the room, started at the floor, and ended at 5 feet height. He identified two ways that vectors are represented notationally to be unique:
1. The <u1, u2, u3> notation (known as u with a tilde '~' accent above) identifies three measurements along three axes, from the 0,0,0 origin, using x, y, and z as typical axes representation. The string could then be identified as a vector by <20, 15, 5> (20 feet on the x (east), 15 feet on the y (north), and 5 feet on the z (upward).
2. The vector could also be represented by its length and an angle in 3-D space (25 feet, 37 degrees to the NE)
which is known as ui notation
note: the 37 degree angle is made up of 48.6 degrees North from East axis, and 14.5 degrees Up from the East axis.
Chris asked students to participate in a speed game whereby each team had to identify a unique vector based on the <u1, u2, u3> notation. For example <15, 15, 0> was represented by:
Chris handed out a homework assignment for students to work on for Friday's upcoming class. To prepare them, he described the dot product between two vectors A and B:
A . B = A1 * B1 * e1.e1 + A2 * B2 * e2.e2 + A3 * B3 * e3.e3 = A * B * cos(angle between them)
(each e#.e# term evaluates to 1)
He said a good reminder of dot product is Jiminy Cricket while the cross product (A x B) is Honest John (to be elaborated upon next class session).
To warm students up to the cos() calculation, Chris mentioned that the cosine of 0 degrees is 1 and the cosine of 90 degrees is 0.
The homework assignment would be continued in class on Friday, including an introduction to using MATLAB in supporting calculation.
