Mathematical Methods

Question 1

What are the two types of coordinates?

Answer 1

Rectangular (cartesian) and polar

Question 2

What is the format of rectangular coordinates?

Answer 2

• Typical (x,y) format in 2D
• (x,y,z) in 3D.
• Sometimes, even more than 3 coordinates might be included in order to represent additional variables such as time and/or temperature (x,y,z,p)

Question 3

What is the format of polar coordinates?

Answer 3

(r,theta)

• r represents radial distance from the origin
• theta represents the angle measured anticlockwise from a line that passes through the origin.
• Polar coordinates are very useful for describing functions that vary continuously and smoothly around a point.

Question 4

What are the two polar coordinate systems used in three dimensions? Describe them.

Answer 4

Cylindrical - An additional axis is added orthogonally to the (r,theta) ones. (r, theta, z) - two distances and one angle.

Spherical - An additional angle is added between the x and the r axis. (r, phi, theta) - one distance and two angles.

Question 5

What does a radian describe? What angle (in degrees) is equal to one radian?

Answer 5

The angle at which the radius and length of the arc are equal.

Pi/180 degrees is equal to one radian.

Question 6

What is the unit for solid angles?

Answer 6

Steradians.

Question 7

What is a vector vs a scalar quantity?

Answer 7

• A vector has both magnitude and direction, whereas a scalar has only magnitude.

Question 8

What is used to represent unit vectors in a 3D vector?

Answer 8

r = Xi + yj + zk.

In this equation, r is the vector, and i j and k all represent the unit vectors. These are one unit long along each of the axes.

Question 9

What is a unit vector?

Answer 9

A vector that has a length of exactly one unit (of whatever the unit being used is).

A unit vector is denoted by putting a hat on it.

Question 10

What is the directional cosine of a vector?

Answer 10

Refers to the cosine value of the angle between a given vector and any one of the axes it is defined by (x, y or z).

A 3D vector will have three directional cosines corresponding to each of the three axes.

Question 11

What is the sum of the squares of all the directional cosines of a vector?

Answer 11

1

Question 12

What are the two ways vectors can be multiplied?

Answer 12

Scalar (dot) product. Gives a scalar quantity. Calculated by:

p.q = |p||q|cos(theta)

(theta is the angle between p and q)

Vector (cross) product. Gives a vector orthogonal to the other two.

p X q = |p||q|sin(theta) ň

(theta is the angle between p and q, ň is the unit vector in an orthogonal direction to p and q - this is the same direction that the cross product vector will be pointing in).

Question 13

What information does the scalar product give us?

Answer 13

• If scalar product is 0, the vectors are perpendicular to one another (because cos (90) is 0, therefore rest of the equation is multiplied by 0).
• Is scalar product is maximum value, the two vectors are parallel (because cos 0 and cos 180 both are 1, therefore giving a maximum value).

Question 14

How does the right hand rule apply to the cross product of vectors?

Answer 14

pointer finger and middle finger point in directions of the original two vectors, then the thumb will point in the direction of the cross product vector.

Question 15

Does the order of vectors in cross product matter?

Answer 15

YES - a X b is not equal to b X a.

However, a X b would be equal to -b X a

Question 16

When is the cross product of two vectors 0?

Answer 16

Cross product is 0 when the two vectors are parallel

Question 17

What is a limit in terms of a function in maths?

Answer 17

A limit refers to the value that a function approaches outputting for a given input value.

For example y = (X^2 - 1)/ (x - 1)

For this function, when x approaches 1 the limit is 2.