University:
University of Michigan
Course:
MATH 215 | Multivariable and Vector Calculus
Academic year:

2023
Views:

121

Pages:

2
Author:

unabilidxx6

Math 215 Examples Space Curves Key Concepts Vector-Valued Functions A vector-valued function is a function that outputs a vector rather than a single number. Often we will be working with functions whose outputs are vectors in three-dimensional space. Such a function can be written as r (⃗ t) = ⟨x(t), y(t), z(t)⟩ where f (t), g(t), and h(t) are usual functions whose outputs are single numbers (sometimes called scalar functions in contrast with vector-valued functions). Space Curves A vector-valued function r (⃗ t) whose values are three-dimensional functions traces out a space curve, a curve in three-dimensional space. For example, r (⃗ t) = ⟨t, t3 t2 18 , 3 ⟩ traces out a space curve called a twisted cubic: Vector-valued functions don't always trace out smooth curves. For example, r (⃗ t) = ⟨ t4 30 , t3 t2 10 , 5 ⟩ ⃗ has a sharp corner at the point r (0) = ⟨0, 0, 0⟩ : Different Parameterizations Note that two different vector-valued functions may trace out the same curve. We say that two such vector-valued functions corresponding to the same curves are different parameterizations of the same curve. Given any vector-valued function r (⃗ t), it is easy to construct different parameterizations of the same curve: just use s (⃗ t) = r (⃗ at) for any non-zero constant a . For example, suppose we have r (⃗ t) s (⃗ t) = ⟨(2 + sin 3t) cos t, (2 + sin 3t) sin t, cos 3t⟩ = ⟨(2 + sin 6t) cos 2t, (2 + sin 6t) sin 2t, cos 6t⟩. ⃗ t) , so as observed above, both these vector-valued functions will trace out the same space curve (a curve called a toroidal spiral): Notice that s (⃗ t) = r (2 Illustrated Example Example Find a vector-valued function that traces out the curve defined as the intersection of the paraboloid z = x2 + y 2 with the plane y = x. Worked Solution Since every equation in this problem is a function of x , we can use x itself as our parameter. In other words, in our eventual parameterization r (⃗ t) = ⟨x (t), y(t), z(t)⟩, we can set x (t) = t . Since our curve lies entirely in the plane y = x , it follows that y(t) = x (t) in our parameterization, or y(t) = t . Since the curve lies on the paraboloid z = x2 + y 2 , we similarly have z(t) = x (t)2 + y(t)2 = 2t2 . Thus our final parameterization of this curve is r (⃗ t) = ⟨t, t, 2t2 ⟩. Visualizing the Example The animation below shows the paraboloid z = x2 + y 2 with r (⃗ t) = ⟨t, t, 2t2 tracing out its intersection with the plane y = x: Further Questions 1. 2. 3. 4. Find a parameterization of the curve in the example that traces out the curve half as fast. Find a parameterization of the curve in the example that traces out the curve backwards. What curve does r (⃗ t) = ⟨t, −t, 2t2 ⟩ trace out? In the animation in Key Concepts with two different parameterizations, which of r (⃗ t) and s (⃗ t) is represented by the blue arrows? Which is represented by the red arrows?

Math 215 Examples Space Curves

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