Chapter 2: Space Curves
Section 2.3: Tangent Vectors
Example 2.3.1
If Rp is the position-vector representation of the parametric curve x=p2, y=p3, p≥0, show that limh→0Rp+h−Rph, denoted by R′p=dRdp, is the vector x′p i+y′p j. Thus, the differentiation operator ddp is applied to R by applying it to each component of R.
Solution
Mathematical Solution
Consider the following calculation:
R′p=dRdp
=limh→0Rp+h−Rph
=limh→01h((p+h)2(p+h)3−p2p3)
=limh→02 p+h3 p2+3 h p+h2
=2 p3 p2
The astute observer will note that in order to demonstrate that the derivative of the position vector is taken componentwise, the limit operator is applied componentwise. However, the componentwise application of the limit seems much more intuitive than that of the derivative. And yes, R′p is the vector whose components are the derivatives of the components of Rp.
Maple Solution - Interactive
Write R=… as per Table 1.1.1.
Context Panel: Assign Name
R=p2,p3→assign
Form the "difference quotient", simplify, and calculate the limit
Expression palette: Evaluation template Press the Enter key.
Context Panel: Simplify≻Simplify
Context Panel: Evaluate at a Point≻h=0
1hRx=a|f(x)p=p+h−R
p+h2−p2hp+h3−p3h
= simplify
h+2ph2+3hp+3p2
→evaluate at point
2p3p2
To obtain the limit of the difference quotient as h→0, h was simply set equal to zero via the Context Panel option "Evaluate at a Point". (See Figure 2.3.1(a).) Since no indeterminate form or division-by-zero error occurs, the resulting expression is the desired limit.
Figure 2.3.1(a) Evaluation at a point
Maple Solution - Coded
Define R as a function of p.
R≔p→p2,p3:
Form the difference quotient DQ.
DQ≔Rp+h−Rp/h
DQ≔p+h2−p2hp+h3−p3h
Use the map command to apply the limit operator to each component of DQ.
maplimit,DQ,h=0 = 2p3p2
Note: The Context Panel option "Assign Function" fails with vectors. Maple uses the arrow to implement the mapping definition of a function, syntax that is hidden by the "Assign Function" option of the Context Panel. The Expression palette contains a template for setting the mapping notation for a function.
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