13.2

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13.2 Derivatives and Integrals

of Vector Functions

Derivatives

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Derivatives

The derivative r′ of a vector function r is defined in much the same way as for real-valued functions:

if this limit exists. The geometric significance of this definition is shown in Figure 1.

Figure 1

(b) The tangent vector r′(t) (a) The secant vector

Derivatives

If the points P and Q have position vectors r(t) and r(t + h), then represents the vector r(t + h) – r(t), which can

therefore be regarded as a secant vector.

If h > 0, the scalar multiple (1/h)(r(t + h) – r(t)) has the

same direction as r(t + h) – r(t). As h → 0, it appears that this vector approaches a vector that lies on the tangent line.

For this reason, the vector r′(t) is called the tangent vector to the curve defined by r at the point P, provided that

r′(t) exists and r′(t) ≠ 0.

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Derivatives

The tangent line to C at P is defined to be the line through P parallel to the tangent vector r′(t).

We will also have occasion to consider the unit tangent vector, which is

Derivatives

The following theorem gives us a convenient method for computing the derivative of a vector function r: just

differentiate each component of r.

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Example 1

(a) Find the derivative of r(t) = (1 + t³)i + te^–t j + sin 2t k.

(b) Find the unit tangent vector at the point where t = 0.

Solution:

(a) According to Theorem 2, we differentiate each component of r:

r′(t) = 3t²i + (1 – t)e^–t j + 2 cos 2t k

Example 1 – Solution

(b) Since r(0) = i and r′(0) = j + 2k, the unit tangent vector at the point (1, 0, 0) is

cont’d

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Derivatives

Just as for real-valued functions, the second derivative of a vector function r is the derivative of r′, that is, r″ = (r′)′.

For instance, the second derivative of the function, r(t) =

〈

〉

r″(t) =

〈

〉

Differentiation Rules

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Differentiation Rules

The next theorem shows that the differentiation formulas for real-valued functions have their counterparts for

vector-valued functions.

Example 4

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

Solution:

Since

r(t)  r(t) = |r(t)|² = c²

and c² is a constant, Formula 4 of Theorem 3 gives

0 = [r(t)  r(t)] = r′(t)  r(t) + r(t)  r′(t) = 2r′(t)  r(t)

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Example 4 – Solution

Geometrically, this result says that if a curve lies on a

sphere with center the origin, then the tangent vector r′(t) is always perpendicular to the position vector r(t). (See

Figure 4.)

cont’d

Figure 4

Integrals

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Integrals

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.

But then we can express the integral of r in terms of the integrals of its component functions f, g, and h as follows.

Integrals

So

This means that we can evaluate an integral of a vector function by integrating each component function.

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Integrals

We can extend the Fundamental Theorem of Calculus to continuous vector functions as follows:

where R is an antiderivative of r, that is, R′(t) = r(t).

We use the notation

∫

Example 5

If r(t) = 2 cos t i + sin t j + 2t k, then

∫

∫

∫

∫

= 2 sin t i – cos t j + t² k + C

where C is a vector constant of integration, and