Advanced Calculus using Mathematica

Advanced Calculus using Mathematica: NoteBook Edition is a complete text on calculus of several variables written in Mathematica NoteBooks.

Parametric 1D: Procedure 9.6.1

A regular parameterization gives a direction or orientation to the curve from the starting point X[a] in the direction of AdvCalcMathWeb_97.png to X[b].

Figure 9.6.2 shows the full-scale figure on the left and a view magnified by 1/δt on the right.  After magnification the red error measures ε.  When δt0 is small, the magnified error ε is small and the magnified displacement from X[t] to X[t+δt] appears to be the same as the linear (in δt) displacement AdvCalcMathWeb_98.png.

Notice that this magnified view depends on the “smooth parameterization” (or “regularity”) condition that  AdvCalcMathWeb_99.png  because otherwise magnification by 1/δt does not reveal the vector AdvCalcMathWeb_100.png from the approximation


The Theorem on Smooth Formulas 3.3.1 is easy to apply, simply verify that the formulas x[t], y[t], z[t], AdvCalcMathWeb_102.png, AdvCalcMathWeb_103.png, AdvCalcMathWeb_104.png are valid in an interval about t to prove smoothness of the functions.  Smoothness of the curve requires the additional condition AdvCalcMathWeb_105.png.  It boils down to: compute and check that your computation makes sense.

Procedure  9.6.1: Finding the Parametric Tangent Line

Suppose AdvCalcMathWeb_106.png is smooth in a neighborhood of a particular fixed t and AdvCalcMathWeb_107.png.    To find the equation of the line tangent to the parametric curve X[t] at t do the following steps:

1.  Compute the general symbolic derivatives AdvCalcMathWeb_108.png, y'[t], AdvCalcMathWeb_109.png and differential,   

AdvCalcMathWeb_110.png  or  AdvCalcMathWeb_111.png

(the symbolic derivatives are functions of  t)

2.  Calculate the specific values of the derivatives at the particular t, AdvCalcMathWeb_112.png, AdvCalcMathWeb_113.png, AdvCalcMathWeb_114.png, and write the specific total differential at this point:

d X=Vd t    or    AdvCalcMathWeb_115.png

The coefficient V is a specific vector or u, v, and w are specific numbers.  This is one vector tangent to the curve at t, provided it is not zero.

3. (Optional)  If you want the equation of the tangent plane in  (x,y,z)-coordinates, say  AdvCalcMathWeb_116.png, AdvCalcMathWeb_117.png, AdvCalcMathWeb_118.png is your particular point of tangency and  AdvCalcMathWeb_119.png, AdvCalcMathWeb_120.png, AdvCalcMathWeb_121.png is your particular velocity. The local  (d x,d y,d z)-coordinates are related to (x,y,z)-coordinates by AdvCalcMathWeb_122.png, AdvCalcMathWeb_123.png, and AdvCalcMathWeb_124.png, so replace  d x,  d y, and d z  with AdvCalcMathWeb_125.png, AdvCalcMathWeb_126.png,  and AdvCalcMathWeb_127.png, obtaining the equation:

AdvCalcMathWeb_128.png or  AdvCalcMathWeb_129.png  or AdvCalcMathWeb_130.png

This procedure is programmed in Section 9.5.

You can plot the tangent line using the equation in local coordinates (d x,d y,d z) centered on the curve at the particular X[t] from Step 2.  The graph of this equation is a parametric line through the (d x,d y,d z)-origin centered at the point X[t].

The result of Step 3 is a parametric line through AdvCalcMathWeb_131.png with constant velocity AdvCalcMathWeb_132.png and parameter d t.  Notice that to correctly evaluate this final expression, you need to perform steps (1), (2), and (3) in that order.

Notice that the procedure breaks down if AdvCalcMathWeb_133.png because the parametric graph of AdvCalcMathWeb_134.png  is just the point AdvCalcMathWeb_135.png and not a line.  We require AdvCalcMathWeb_136.png in order to know that the curve C that X[t] parameterizes is geometrically smooth. (See the non-smooth parametric curve below.)  This requirement guarantees that the procedure produces a line and magnification shows that it is tangent.


Find the velocity vector and tangent line to the helix


at the point where t=π/6.


Step 1.  The general symbolic differential is

AdvCalcMathWeb_138.png  or derivative AdvCalcMathWeb_139.png

Since all the coordinate functions and all their derivatives are defined everywhere, the function is smooth.  Since AdvCalcMathWeb_140.png, the graph is smooth.

Step 2.  At the particular point where  t=π/6,

AdvCalcMathWeb_141.png  or  AdvCalcMathWeb_142.png

This is the equation of the line through the local AdvCalcMathWeb_143.png origin with constant velocity AdvCalcMathWeb_144.png.  

We place local coordinates at the point AdvCalcMathWeb_145.png, AdvCalcMathWeb_146.png, AdvCalcMathWeb_147.png, draw the constant velocity vector AdvCalcMathWeb_148.png, and sketch the line through it:


Figure: Tangent to a helix