maths B

Question 1

What is the del operator?

Answer 1

Question 2

What is grad?

Answer 2

Question 3

What is the div operator?

Answer 3

the answer should be a scalar

Question 4

What is the curl operator?

Answer 4

Question 5

What is a scalar field?

Answer 5

A function that takes 2 or three values in and outputs one.

Question 6

What is a vector field?

Answer 6

That is to say you put three values in and you get three values out

Question 7

What does |a| and ||a|| denote?

Answer 7

|| || denotes the magnitude of a bound vector. That is to say a vector with an origin.

| denotes the magnitude of a free vector

Question 8

What is the dot product equal to in terms of the angle between vectors?

Answer 8

A.B=|A| |B| cosα

Question 9

How can you define the cross product?

Answer 9

AxB=|A| |B| n sinα

n is a unit vector perpendicular to both a and b pointing in the direction defined by the right hand rule.

Question 10

What is the definition of the line integral of a scalar field?

Answer 10

Question 11

How do you find the line integral of a vector field?

Answer 11

Question 12

How can you determine whether a vector field is conservative?

Answer 12

if curl (f)=0 it is conservative

Question 13

How do you find the scalar potential of a conservative vector field, f?

Answer 13

grad(φ)=f

Question 14

How do you find the work done moving from one point to another when you have the scalar potential?

Answer 14

Calculate the potential at the two different points and the difference is the work

Question 15

What is the laplace operator?

Answer 15

Δ=div grad

∇.∇=∇²=Δ

There is also the vector laplacian

Question 16

If you have the cyclindrical co-ordiantes, ρ,Φ,z, how would you express it in cartesian co-ordinates?

Answer 16

r=ρcosΦi + ρsinΦj + zk

Question 17

What is the formula of integration by parts?

Answer 17

Question 18

In 3D, what is the general form from going to one set of basis to another?

Answer 18

Question 19

If you have three vectors, how do you know if they are right handed?

Answer 19

You cross product two of them together to make the third

Question 20

How do you evaluate a surfce integral of the first kind?

Answer 20

The first kind means you are integrating within a scalar field

Question 21

How do you evaluate the surface integral of the second kind?

Answer 21

Question 22

What is the parametic equation of a sphere radius r, centred at the origin?

Answer 22

R*cos(t)sin(s), R*sin(t)sin(s), R*cos(s)

t betweeb 0 and 2π

s between 0 and π

Question 23

What is a positively oriented curve?

Answer 23

A positively oriented curve is a closed piecewise smooth simple (non-intersecting) curve whose interior is on the left as we traverse the curve.

This means that, in order to have a positively oriented curve, a particular parametrization r(t) : [a, b] has to be chosen so that starting at t = a, we travel, roughly speaking, in the counter-clockwise direction around the curve as t increases from a to b.

Question 24

What is stoke’s theorem?

Answer 24

Question 25

What is the Gauss divergence theorem?

Answer 25

The triple inegral is the exprestion for the volume of the shape.

Question 26

What is grad of a vector field?

Answer 26