Real Functions of Several Variables & Partial Derivatives Flashcards
What is the special case of a graph where there are 2 real variables called?
Its surface.
What is the difference between a dependant and independant variable?
A dependant variable relies on independant variables.
For a function f of n variables, what are the names for special cases of its level set?
n = 2 - level curve
n = 3 - level surface
n > 3 - level hypersurface
Can the xy plane belong to more than 1 level set?
No.
What idea is the continuity of function f dependant on?
The continuity of f depends on its limit.
What is the symbol for differentiation of more than one variable?
Delta - ∂.
In order to derive a function from partial derivatives, what must we introduce?
An arbitrary function of all variables not included in the differentiation. This extends beyond the arbitrary constant in integration of single variabled functions.
If a function f(x, y) satisfies Laplaces Equation, what does this mean?
∂2f/∂x^2 + ∂^2f/∂y^2 = 0.
If a function f(x, t) satisfies the Heat Equation, what does this mean?
∂f/∂t = ∂^2f/∂X^2.
If a function f(x, t) satisfies the Wave Equation, what does this mean?
∂^f/∂t^2 = c^2 . ∂^2f/∂x^2 for any c.
What is the equation for the gradient of f(x, y)?
∇f(x, y) ≡ gradf(x, y) = ∂f/∂x(x, y) e1 +∂f/∂y (x, y) e2.
