21-259: Calculus in Three Dimensions

Lecture #5

Spring 2025

Vector Functions and Space Curves

Definition: A vector function r (t) : R→R n is a function whose domain is a set of real numbers and whose range is a set of vectors.

For n = 3, r (t) = = f (t)ı + g (t) ȷ +h(t)k is a vector function. The scalar functions f (t), g (t), and h(t) are component functions of r (t).

Example 1. Find the domain of the vector function r (t) = ­ <3 ,ln(3− t), √t>.

If r (t) = ­, then

provided the limits of the component functions exist.

Example 2. Find

A vector function r (t) is continuous at t = a if r (t) = r (a).

Definition: Let f , g , and h be continuous functions on an interval I. Then the set C of all points (x, y, z) in space, where

x = f (t),      y = g (t),      z = h(t),

is called a space curve. The equations above are the parametric equations of C, and t is called a parameter. Any continuous vector function r (t) defines a space curve.

Example 3. What are the parametric equations of a circle of radius a in the x y-plane, centered at the origin?

Example 4. What is the curve given by r (t) = cos tı +sint ȷ + tk?

Example 5. Find a vector function that represents the curve of intersection of the cylinder x2 + y2 = 1 and the plane y + z = 2.

Example 6. Find a vector function that represents the curve of intersection of the paraboloid z = 4x2 + y2 and the parabolic cylinder y = x 2 .

Calculus of Vector Functions

Definition: The derivative r′ of a vector function r is given by

provided the limit exists.

For any value of t, the vector r′ (t) is the tangent vector to the curve defined by r, provided that r′ (t) exists and r′ (t) ≠ 0. The vector

is the unit tangent vector to r (t).

Theorem: If r (t) = ­ = f (t)ı + g (t) ȷ + h(t)k, where f , g , and h are differentiable functions, then

r′ (t) = ­<f′ (t), g′ (t),h′ (t)> = f′ (t)ı + g′ (t) ȷ +h′ (t)k.

Example 7. For the vector function r (t) = ­ , find r′ (t), and find T (0).

Example 8. Find parametric equations for the tangent line to the curve x = lnt, y = 2√t, z = t 2 at the point (0,2,1).

Theorem: Suppose u and v are differentiable vector functions, c is a scalar, and f is a real-valued function. Then

1. dt/d (u(t)+ v(t)) = u ′ (t)+ v ′ (t)

2. dt/d  (cu(t)) = cu′ (t)

3. dt/d (f (t)u(t)) = f (t)u′ (t)+ f′ (t)u(t)

4. dt/d (u(t)· v(t)) = u(t)· v′ (t)+u′ (t)· v(t)

5. dt/d (u(t)× v(t)) = u(t)× v′ (t)+u′ (t)× v(t)

6. dt/d (u(f (t))) = f′ (t)u′ (f (t))

Example 9. If r (t) ≠ 0, show that

Example 10. Show that if |r (t)| = c (where c is a nonzero constant) then r′ (t) is orthogonal to r (t) for all t.

Example 11. Show that if r is a vector function such that r′′ exists, then

Definition: The definite integral of a continuous vector function r (t) = ­ can be defined in much the same way as for real-valued functions except that the integral is a vector:

Example 12. Evaluate