240 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION This result ensures that functions deFned by algebraic or analytic expressions such as polynomials (in two or three variables) or combinations of trigonometric, exponential, logarithmic functions and roots are generally di±erentiable, since by the formal rules of di±erentiation the partials are again of this type, and hence are continuous wherever they are deFned; the only di²culties arise in cases where di±erentiation introduces a denominator which becomes zero at the point in question. The Gradient and Directional Derivatives Recall from § 3.2 that a linear function can be viewed in three di±erent ways: as a homogeneous polynomial of degree one, as multiplication of the coordinate matrix by its matrix representative , and as the dot product of the input with a Fxed vector. We have seen that when f ( −→ x ) is di±erentiable at −→ x = −→ x0 , then the coe²cients of the di±erential d −→ x0 f ( −→ v ), as a polynomial in the entries of
This is the end of the preview. Sign up
access the rest of the document.
This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.