Chapter 1-Curves&Tangents Flashcards
Define a curve.(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.
Is the modulus scalar or vector?(1)
scalar
Is the unit vector scalar or vector?(1)
vector
What does r(t) represent?(1)
The position vector with coordinates x and y.
Define a parabola in terms of x and y.(1)
y=x^2
Whats interesting about the parametric form of a curve?(1)
Not unique, a curve can be parameterised in infinite ways
What is the explicit geometric representation of the curve?(1)
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
Give sin values for sin(x) where x=
0, pi/2, pi/ 3pi/2, 2pi.
Cos for same cos(x).(2)
Sin 0, 1, 0, -1, 0
Cos 1, 0, -1, 0, 1
What does it mean if something is piecewise smooth?(1)
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.
What is the displacement vector?(1)
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)
What is the velocity vector?(1)
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)
Define the line element.(1)
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.
How to find the tangent to the curve if given in parametric form?(1)
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.
When are dummy integration variables used?(1)
If using the variable as the range for the integration, cannot also integrate so t goes to t’.
What is arc length?(1)
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.
Define a scalar field.(1)
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.
cosh^2(t)=?
1+sinh^2(t)
sinh(t)=?
1/2(e^t-e^-t)
Does the length of the curve depend on its parameterisation?(1)
No, tangent does not actual curve length
tangent is a vector parallel to the curve at a particular part.
Define the vector field.(1)
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.
What is the line integral for a vector field?(1)
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.
If youre trying to parameterise and struggling to get the right direction what can you do to help?(1)
Use minus t instead of t.
Define a surface.(1)
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.
