文档介绍：Department of Mathematics
Semester 2 2010/2011
MA1104 Multivariable Calculus
– Chapter 2 –
Vector Functions of One Variable
Contents:
Calculus of Vector Functions
Antiderivative and Integral
Tangent Vector and Tangent Line to a Curve
Smooth Curves
Arc Length
2 Vector Functions of One Variable
Sometimes ideas e to me. Other times I have to sweat and bleed
to make e. It’s a mysterious process, but I hope I never find out
exactly how it works.
– J. K. Rowling
Recall the vector equation of line:
r(t) = r0 + tv.
We have seen that the tip of the vector r(t) traces out a line as t varies.
We can rewrite the above as follows:
h i h i h i
r(t) = x0,y0,z0 + t a,b,c = x0 + ta,y0 + tb,z0 + tc .
Notice that ponent of r(t) is a scalar function of t.
In general, a vector-valued function is
r(t) = hf (t),g(t),h(t)i
where f (t), g(t) and h(t) are scalar functions of t.
Formally,
⊆⊆
A vector-valued function r(t) is a mapping from its domain D R to its range R V3
(the set of all vectors in space), so that for each t ∈ D, r(t) = v for exactly one vector
∈
v V3.
We write a vector-valued function as
r(t) = f (t)i + g(t)j + h(t)k
1
or
r(t) = hf (t),g(t),h(t)i
for some scalar function f , g and h (called ponent functions of r).
Note that r(t) traces out a curve in space!
If C is that curve, we say that r(t) is a parametrization of C.
Conversely, a curve C can have more than one parameterizations.
For example, both
r(t) = ht,t2i, t ∈ R
r(t) = ht3,t6i, t ∈ R
parameterize the parabola f (x) = x2 on the xy-plane.
Example Sketch the curve traced out by the vector-valued function r(t) = sinti −
3costj + 2tk.
2
Solution. There is a relationship between x and y here:
y 2
x2 + = sin2 t + cos2 t = 1
3
which is the equation of an ellipse in 2-D. In 3-D, since the equation does not involve
z, it es the equation of an elliptic cylinder whose axis is the z-axis.
The curve will wind its way up the cylin