Derivatives and Integrals of Vector Functions

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Section 14.2

• Derivatives and Integrals of Vector Functions

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THE DERIVATIVE OF VECTOR FUNCTION
The derivative r'(t) of a vector function r is
defined to be provided the limit exists.
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TANGENT VECTORS AND TANGENT LINES

• If r(t0) is the position vector for the point
  P0(x0, y0, z0), then the vector r'(t0) is called
  the tangent vector to the curve defined by r and
  the point P0, provided that r(t0) exists and
  r(t0) ? 0.
• The tangent line to C at P0 is defined to be the
  line through P0 parallel to the tangent vector
  r(t0).
• The unit tangent vector is defined to be

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THEOREM
If
, where f, g, and h are
differentiable functions, then
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SMOOTH CURVES
A curve given by a vector function r(t) on an
interval I is called smooth if r' is continuous
and r' ? 0 (except possibly at the endpoints of
I).
A curve is called piecewise smooth if it is made
up of a finite number of smooth pieces.
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DIFFERENTIATION RULES
Suppose u and v are differentiable vector
functions, c is a scalar, and f is a real-valued
function. Then
(Chain Rule)
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THE DEFINITE INTEGRAL
The definite integral of a continuous vector
function r(t) can be defined in the same way as
for real-valued functions. This will simplify to
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THE FUNDAMENTAL THEOREM OF CALCULUS AND VECTOR
FUNCTIONS
Let R be an antiderivative of r, that is, R'(t)
r(t). Then
NOTE We use the notation ?r(t)dt for
indefinite integrals (antiderivatives).