Chapter 1-Curves&Tangents

Question 1

Define a curve.(1)

Answer 1

A single continuous curve,C, is a set of points specified by two continuous functions, x(t) and y(t), of a single variable t, for some range t1<=t<=t2 (here, t1 and t2 are constants).
Pure maths version: Mapping of single real variable t wiht specific range to variables x and y so mapping R to R^2
t=curve parameter.
Natural then to talk about direction and speed as think of point r(t) moving through the curve through t as in time.

Question 2

Is the modulus scalar or vector?(1)

Answer 2

scalar

Question 3

Is the unit vector scalar or vector?(1)

Answer 3

vector

Question 4

What does r(t) represent?(1)

Answer 4

The position vector with coordinates x and y.

Question 5

Define a parabola in terms of x and y.(1)

Answer 5

y=x^2

Question 6

Whats interesting about the parametric form of a curve?(1)

Answer 6

Not unique, a curve can be parameterised in infinite ways

Question 7

What is the explicit geometric representation of the curve?(1)

Answer 7

If at least one of the functions x=x(t)or y=y(t) can be inverted (e.g. if we can write t=t(x)) then we can substitute into the other function and thus eliminate t. E.g. if y=y(t) and t=t(x) then y=y(t(x))≡f(x). E.g. in the Example we considered earlier, with x=t and y=t^2, we could simply write t=x, and hence y=x^2.
Note: More generally, eliminating t might result in an implicit geometric representation of the formF(x, y)=0

Question 8

Give sin values for sin(x) where x=
0, pi/2, pi/ 3pi/2, 2pi.

Cos for same cos(x).(2)

Answer 8

Sin 0, 1, 0, -1, 0

Cos 1, 0, -1, 0, 1

Question 9

What does it mean if something is piecewise smooth?(1)

Answer 9

Can be split into smaller smooth pieces eg square can be split into 4 “smooth” parts, useful for sectioning things that are harder to parametrise when given in implicit form.

Question 10

What is the displacement vector?(1)

Answer 10

Thinking of t as in time, as r(t) moves along the curve by delta, the displacement is the direction of the vector from r(t) to the r(t+delta) ie r(t+delta)-r(t)

Question 11

What is the velocity vector?(1)

Answer 11

Essentially derivative of position vector:
v= limδt→0(δx/δt,δy/δt)=(dx/dt,dy/dt)=dr/dt

Is a tangent to the curve! Denoted v(t)

Question 12

Define the line element.(1)

Answer 12

The quantity dr=v dt is called the line element of the curve. It is an infinitesimal tangent vector of magnitude dr=|dr|=|(dx,dy)|=√(dx^2+d^2 is the modulus.

Question 13

How to find the tangent to the curve if given in parametric form?(1)

Answer 13

Differentiate with t for x and y and then if t value given sub in.

Remember tangents can have different magnitude due to diff parameterisations BUT will all have same direction.

Question 14

When are dummy integration variables used?(1)

Answer 14

If using the variable as the range for the integration, cannot also integrate so t goes to t’.

Question 15

What is arc length?(1)

Answer 15

We can also define the length (fromt1)toanarbitrary intermediate point on the curve. This is the arc-length or partial length s(t),

Given by the integral between t and t1 of v(t’) dt’ where t’ is a dummy integral.

Question 16

Define a scalar field.(1)

Answer 16

Pure maths definition would be a mapping from R^2 to R, ie temperature in a bath, pick a point in the bath and get the temperature answer. ie (x,y)–>f(x,y) or f(r)

use of f(r) is generalises to higher dimensions.

Question 17

cosh^2(t)=?

Answer 17

1+sinh^2(t)

Question 18

sinh(t)=?

Answer 18

1/2(e^t-e^-t)

Question 19

Does the length of the curve depend on its parameterisation?(1)

Answer 19

No, tangent does not actual curve length

tangent is a vector parallel to the curve at a particular part.

Question 20

Define the vector field.(1)

Answer 20

A vector quantity defined at a specific point a 2D vector field woule be a mapping from R^2 to R^2, a real life example would be the direction of airflow in a room, each point has a magnitude and direction but may well differ for different parts of the room.
Denotes as (x,y)--> F(x,y)--->(Fx(x,y), Fy(x,y))

Note Fx and Fy are functions of coordinates so are actually scalar fields.

Question 21

What is the line integral for a vector field?(1)

Answer 21

It is scalar!! due to its dot proudct nature

It is the integral of (Fx,Fy) dot dr ie (dx,dy) the Line element on C and is a VECTOR.

Question 22

If youre trying to parameterise and struggling to get the right direction what can you do to help?(1)

Answer 22

Use minus t instead of t.

Question 23

Define a surface.(1)

Answer 23

A single surface,S, in 3D space (x, y, z) is a set of points specified by three continuous functions of two variables (u, v), say.