Tangent vectors, or Copyright 2008 by Evans M. Harrell II. MATH Harrell

Tangent vectors, or how to go straight when you are on a bender. Copyright 2008 by Evans M. Harrell II. MATH 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) = t 2 and f(x) = sin(x)i - 2 x j,  Example: If u(t) = t 2 and f(x) = sin(x)i - 2 x j, then f(u(t)) = sin(t 2 )i - 2 t 2 j, and its derivative:  2 t cos(t 2 )i - 4 t j is equal to  2 t (cos(x)i - 2j) when we substitute x = t 2.  2 ways to calculate: substitute and then differentiate, or chain rule

 circles and ellipses  spirals  helix  Lissajous 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).Rogness applets (UMN).

Tangent vectors  Think velocity!  Tangent lines  Approximation and Taylor’s formula

Tangent vectors  Think velocity!  Tangent lines  Approximation and Taylor’s formula  Numerical integration

Tangent vectors  Think velocity!  Tangent lines  Approximation and Taylor’s formula  Numerical integration  Maybe 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: 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

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

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

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)| dt  In 2-D ds = (1 + y’ 2 ) 1/2 dx, or ds 2 = dx 2 + dy 2 or…

Arc length

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