By H.J. de Vega, N. Sanchez
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Extra info for Field Theory, Quantum Gravity and Strings
Sample text
Cn , with cn = 0 are called the coefficients. This set is closed under addition and complex scalar multiplication: the sum of two polynomials with complex coefficients is itself a polynomial with complex coefficients; likewise the product of a complex number and a polynomial with complex coefficients is again a polynomial with complex coefficients. If we consider polynomials with real coefficients, we get a real vector space that is not a complex vector space: it is not closed under multiplication by complex scalars.
A function f is surjective (onto its target space T ) if the image is equal to the target space. The preimage (under f ) of a subset U of the target space T , denoted f −1 [U ], is the set of all s in the domain of f such that f (s) ∈ U ; in other words, f −1 [U ] := {s ∈ S : f (s) ∈ U } . Similarly, the image (under f ) of a subset U of the domain is the set f [U ] := { f (u) : u ∈ U }. We will often define functions in terms of other functions. For example, The composition of two functions f and g is the function f ◦ g: f −1 [domain of g] → target space of f x → f (g(x)).
This technique, which we will apply to Laplace’s equation, is a method physicists use to find solutions to many differential equations. The technique is often successful, so physicists tend to keep it in the top drawer of their toolbox. 3. , an equation involving derivatives with respect to more than one independent variable), one can often construct some solutions by looking for solutions that are the product of functions of one variable. We will apply this technique, called separation of variables, to find harmonic functions of three variables.