Multivariable Calculus: Vector-Valued Functions flashcards

Question 1

What form does a vector-valued function in three dimensions take?

Accepted Answer

r(t) = f(t)i + g(t)j + h(t)k, where f, g and h are real-valued functions of the parameter t.

Question 2

What is the graph of r(t) = f(t)i + g(t)j called?

Accepted Answer

A plane curve.

Question 3

What is the graph of r(t) = f(t)i + g(t)j + h(t)k called?

Accepted Answer

A space curve.

Question 4

How do you find the limit of a vector-valued function?

Accepted Answer

Take the limit of each component function separately.

Question 5

What is the limit definition of the derivative r'(t)?

Accepted Answer

r'(t) = lim as Δt -> 0 of [r(t + Δt) - r(t)]/Δt, provided the limit exists.

Question 6

How do you differentiate a vector-valued function?

Accepted Answer

Differentiate each component function and use the results as the components of the new function.

Question 7

How is the unit tangent vector T(t) found from r(t)?

Accepted Answer

Divide the derivative by its magnitude: T(t) = r'(t)/||r'(t)||.

Question 8

What is the curvature of a circle of radius r?

Accepted Answer

1/r, the reciprocal of its radius.

Question 9

What is the radius of curvature?

Accepted Answer

The reciprocal of the curvature.

Question 10

What is the principal unit normal vector N(t)?

Accepted Answer

N(t) = T'(t)/||T'(t)||

Question 11

How is the binormal vector B(t) defined?

Accepted Answer

B(t) = T(t) × N(t)

Question 12

Which three vectors form the Frenet (TNB) frame?

Accepted Answer

The unit tangent vector T, the principal unit normal vector N and the binormal vector B.

Question 13

What is the osculating circle of a curve at a point?

Accepted Answer

The circle tangent to the curve at that point that has the same curvature as the curve there.

Question 14

Which two vectors determine the osculating plane?

Accepted Answer

The unit tangent vector T and the unit normal vector N.

Question 15

What is the normal plane to a curve at a point?

Accepted Answer

The plane perpendicular to the curve at that point.

Question 16

If r(t) is the position of an object, what does its first derivative r'(t) represent?

Accepted Answer

Its velocity at time t.

Question 17

If r(t) is the position of an object, what does its second derivative r''(t) represent?

Accepted Answer

Its acceleration at time t.

Question 18

What is the magnitude of the velocity vector called?

Accepted Answer

Speed.

Question 19

What are the tangential and normal components of acceleration?

Accepted Answer

The coefficients of T and N when acceleration is expressed as a linear combination of T and N.

Question 20

What is projectile motion?

Accepted Answer

Motion of an object given an initial velocity, with gravity as the only force acting on it.

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