Problem #29, Section 13.5 (p. 589)

The problem asks us to find the unit normal vector using the formula N = d T /dt / |d T /dt| for the ellipse that corresponds to the position vector

> r := [ a*cos(omega*t), b*sin(omega*t) ];

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The velocity vector is the derivative of the position vector

> v := diff( r, t );

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and the speed is the magnitude of the velocity vector

> s := sqrt( v[1]^2 + v[2]^2 );

which simplifies to

> s2 := simplify( s, assume=positive );

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The unit tangent vector is obtained by dividing the velocity by the speed

> T := [ v[1]/s2, v[2]/s2 ];

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The unit normal vector is the unit vector in the direction of the derivative of the unit tangent vector

> Tp := diff( T, t );

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> Tp_len := sqrt( simplify(Tp[1]^2 + Tp[2]^2) );

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> N := Tp / Tp_len;

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> N2 := [ Tp[1]/Tp_len, Tp[2]/Tp_len ];

> N3 := simplify( % );

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Do not be concerned about the complicated form for this vector. That's the way it turns out for this problem. The point, however, is that the same process can be used for ANY problem in which the position vector is provided.

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To conclude, let's check that these vectors are feasible. This will be done by checking that T and N have unit length and that T and N are orthogonal.

|T| = |N| = 1:

> simplify( T[1]^2 + T[2]^2 );

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> simplify( N3[1]^2 + N3[2]^2 );

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T . N = 0

> simplify( T[1]*N3[1] + T[2]*N3[2] );

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Good News!