For a function f(x, y), the limit exists if the function approaches a specific value as (x, y) approaches a point (a, b) from all possible directions. Mathematically,
lim (x, y) → (a, b) f(x, y) = L
Continuity means that the function is defined at (a, b), the limit exists, and the function value equals the limit.
Differentiability refers to the existence of partial derivatives and whether the function can be approximated by a linear function in the neighborhood of the point.
The chain rule for functions of multiple variables is used to differentiate composite functions. If z = f(x, y), where x and y are functions of t, then:
dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)
Changing variables in multivariable functions helps simplify problems. Commonly used transformations include polar, cylindrical, and spherical coordinates. For example:
x = r cos(θ), y = r sin(θ) in polar coordinates.
If a function f(x, y) is homogeneous of degree n, then it satisfies Euler’s theorem:
x(∂f/∂x) + y(∂f/∂y) = nf(x, y)
The Jacobian is a determinant that represents the rate of change of variables in a multivariable function. For functions x = g(u, v) and y = h(u, v), the Jacobian is given by:
J = | ∂(x, y)/∂(u, v) | = (∂x/∂u ∂x/∂v) / (∂y/∂u ∂y/∂v)
To find the local maxima, minima, or saddle points of f(x, y), we solve:
∂f/∂x = 0 and ∂f/∂y = 0
Then, use the second derivative test by evaluating the discriminant:
D = fxxfyy - (fxy)²
This method is used to find extrema of functions subject to constraints. If we want to maximize or minimize f(x, y) subject to the constraint g(x, y) = 0, we solve:
∇f = λ∇g
Where λ is the Lagrange multiplier. This generates a system of equations that helps find the constrained extrema.