Calculus Volume 3 · free preview
Calculus of Vector-Valued Functions
Vector-valued functions differentiate and integrate componentwise, the derivative being velocity and the integral the displacement, with differentiation rules mirroring the scalar product and chain rules.
1. The derivative as tangent
The derivative as tangent
2. The integral and the fundamental theorem
The integral and the fundamental theorem
3. Differentiation rules in vector form
Differentiation rules in vector form
4. Acceleration and the derivative hierarchy
Acceleration and the derivative hierarchy
5. Componentwise derivative
The derivative of r(t) = (f(t), g(t), h(t)) equals which vector?
6. Tangent line construction
The tangent line to the curve r(t) at t = a is which vector expression?
7. Dot product rule
The derivative of u dot v equals which expression?
8. Constant magnitude
A curve r(t) with constant magnitude has which property?
9. Position from velocity
A particle's velocity is v(t) and its position at time 0 is r_0. Its position at time t is which expression?
10. Fundamental theorem vector form
The integral from a to b of r prime of t dt equals r(b) minus what vector? Answer like r(a).
11. Cross product rule order
In (u cross v) prime = u prime cross v + u cross v prime, the order of factors in each term must be what? Answer with one word.
12. Constant vector of integration
The indefinite integral of r(t) adds which kind of constant? Answer with two words.
13. Chain rule composition
The derivative of r(g(t)) with respect to t equals which expression?
14. Smoothness hypothesis
The arc length formula requires r prime nonzero, the smoothness hypothesis excluding which features? Answer with one word.
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