Section 1 - Radford University
Section 10.2: Derivatives and Integrals of Vector Functions
Practice HW from Stewart Textbook (not to hand in)
p. 707 # 3-21 odd, 29-35
Differentiation of Vector Functions
Differentiation of vector valued functions are done component wise in the natural way. Thus, for
1. 2D Case: If r(t ) = f (t) i + g (t) j, then [pic]
2. 3D Case: If r(t ) = f (t) i + g (t) j + h (t) k , then [pic]
Example 1: Find the derivative of the vector function
r(t ) =[pic].
Solution:
█
Example 2: Find the derivative of the vector function
r(t ) = [pic] i + [pic] j + [pic] k
Solution:
█
Note: Look at properties involving the derivative of vector value functions on p. 705 Theorem 3 of Stewart text.
Tangent Vector to a Vector Valued Function
Recall that the derivative provides the tool for finding the tangent line to a curve. This same idea can be used to find a vector tangent to a curve at a point. We illustrate this idea in the following example.
Example 3: For the vector function r(t ) = [pic] i + [pic] j,
a. Sketch the plane curve with the given vector equation.
b. Find [pic]
c. Sketch the positive vector r(t ) and the tangent vector [pic] at t = 0.
Solution:
█
Example 4: Find the parametric equations for the tangent line to the curve with the parametric equations [pic] at the point (-1, 1, 1).
Solution:
█
Note: Sometimes, it is convenient the normalize a vector tangent to a vector valued function. This gives the unit tangent vector.
Unit Tangent Vector
Given a vector function r on an open interval I, the unit tangent vector T(t) is given by
[pic] where [pic]
Example 5: Find the unit tangent vector T(t) for r(t ) = [pic] i + [pic] j + [pic] k at [pic].
Solution: The unit tangent vector is given by the formula
[pic].
Using r(t ) = [pic] i + [pic] j + [pic] k, we see that
[pic]
and
[pic]
Thus,
[pic]
At [pic], we have (continued on next page)
[pic]
The following graph plots in 3D space the vector function r(t) and the corresponding unit tangent vector T(t) evaluated at [pic].
[pic]
█
Integrals of Vector Functions
Integrals of Vector Valued Functions are computed component wise.
1. 2D Case: If r(t ) = f (t) i + g (t) j, then [pic]
2. 3D Case: If r(t ) = f (t) i + g (t) j + h (t) k , then [pic]
Example 6: Evaluate the integral [pic]
Solution:
█
Example 7: Evaluate the integral [pic]
Solution:
█
Example 8: Find r(t) if [pic] and r(0) = i + j.
Solution: Writing [pic], we see that
[pic]
Thus, [pic] and we need to use the initial condition r(0) = i + j to find the constant vector C. We see that
[pic]
which gives C = i + j. Thus, substituting for C gives
[pic],
which, when combining like terms, gives the result.
[pic]
█[pic]
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