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Tangent vectors, or .......
Copyright 2008 by Evans M. Harrell II.
MATH 2401 - Harrell
Tangent vectors, or how to go straight when you are on a
bender.
Copyright 2008 by Evans M. Harrell II.
MATH 2401 - Harrell
In our previous episode:
1. Vector functions are curves. The algebraic side of the mathematian’s brain thinks about vector functions. The geometric side sees curves.
In our previous episode:
1. Vector functions are curves.
2. Don’t worry about the basic rules of calculus for vector functions. They are pretty much like the ones you know and love.
The good news: The rules of vector calculus look just like the
rules of scalar calculus Chain rule
(f(u(t))) = u(t) f(u(t))
Example: If u(t) = t2 and f(x) = sin(x)i - 2 x j , Example: If u(t) = t2 and f(x) = sin(x)i - 2 x j ,
then f(u(t)) = sin(t2)i - 2 t2 j , and its derivative:
2 t cos(t2)i - 4 t j is equal to2 t (cos(x)i - 2j) when we substitute x = t2 .
2 ways to calculate: substitute and then differentiate, or chain rule
circles and ellipsesspiralshelixLissajous figures
Some great curves and how to write them as parametrized curves
Tangent vectors – the derivative of a vector
function
Tangent vectors
Think velocity!
Tangent vectors
Think velocity!
Tangent vectors
Think velocity!
Tangent vectors
Think velocity!
Tangent vectors
Think velocity!Tangent lines
Tangent vectors
Think velocity!Tangent lines
Tell us more about these!
Tangent vectors
The velocity vector v(t) = r(t) is tangent to the curve – points along it and not across it.
Example: spiral
Curves
Position, velocity, tangent lines and all that: See Rogness applets (UMN).
Tangent vectors
Think velocity!Tangent linesApproximation and Taylor’s
formula
Tangent vectors
Think velocity!Tangent linesApproximation and Taylor’s
formulaNumerical integration
Tangent vectors
Think velocity!Tangent linesApproximation and Taylor’s
formulaNumerical integrationMaybe most importantly – T is our
tool for taking curves apart and understanding their geometry.
Example: helix
Example: Helix
Unit tangent vectors
Move on curve with speed 1.
Unit tangent vectors
Move on curve with speed 1.
T(t) = r(t)/ |r(t)|
Unit tangent vectors
Move on curve with speed 1.
T(t) = r(t)/ |r(t)|
Only 2 possibilities, ± T.
Example: ellipse
Example: ellipse
Example: ellipse
Example: spiral
Example: spiral
Example: Helix
Curves
Even if you are twisted, you have a normal!
Normal vectors
A normal vector points in the direction the curve is bending.
It is always perpendicular to T.
What’s the formula?……………
Normal vectors
N = T/||T||.
Unless the curve is straight at position P, by this definition N is a unit vector perpendicular to T. Why?
Example: spiral in 3D
Example: spiral in 3D
Example: spiral in 3D
Interpretation. Notice that the normal vector
has no vertical component. This is because the spiral lies completely in the x-y plane, so an object moving on it is not accelerated vertically.
Example: Helix
Example: Helix
Example: Helix
Interpretation. Notice that the normal vector again has no vertical component. If a particle rises in a standard helical path, it does not accelerate upwards of downwards. The acceleration points inwards in the x-y plane. It points towards the central axis of the helix.
Example: solenoid
Example: solenoid
Scary, but it might be fun to work it out!
Arc length
If an ant crawls at 1 cm/sec along a curve, the time it takes from a to b is the arc length from a to b.
Arc length
If an ant crawls at 1 cm/sec along a curve, the time it takes from a to b is the arc length from a to b.
More generally, ds = |v(t)| dt
Arc length
If an ant crawls at 1 cm/sec along a curve, the time it takes from a to b is the arc length from a to b.
More generally, ds = |v(t)| dtIn 2-D ds = (1 + y’2)1/2 dx, or ds2 = dx2 + dy2
or…
Arc length
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