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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