Functions that take multiple arguments: f(x_1, x_2, \cdots, x_n) = y
let f(x, y) = z then for some constant c, f(x, y) = c is called a level curve
Finding a limit of a multivariate functions is much more difficult (even when there are only 2 arguments) because instead of approaching a point from left or right we have infinite ways of approaching it.
\lim_{(x, y) \to (x_0, y_0)} f(x, y) = L
If the limit is different when approaching the point from 2 different paths then we can say the limit does not exist.
Example: let f(x, y) = \frac{xy}{x^2 + y^2} when (x, y) \to (0, 0)
Let \lim_{(x, y) \to (x_0, y_0)} f(x, y) = L and \lim_{(x, y) \to (x_0, y_0)} g(x, y) = M
Sandwich theorem still applies.
A function is considered continuos if \lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0)
Derivatives of a function with respect to a given argument.
\frac{\partial f}{\partial x} (x_0, y_0) \equiv \frac{d}{dx} f(x, y_0)|_{x=x_0} \equiv f_x(x_0, y_0) = \lim_{h \to 0} \frac{f(x_0 + h, y_0) - f(x_0, y_0)}{h}
We call a partial derivative pure when all partial derivatives were taken with respect to the same variable. Example: f_{xx} or f_{yy}
We call a partial derivative mixed when partial derivatives were taken with respect to different variables. Example: f_{xy} or f_{yx}
If a function and its derivatives (f_x, f_y, f_{yx}, f_{xy}) are defined and continuous on an open region containing a point (x_0 ,y_0) then f_{xy}(x_0, y_0) = f_{yx}(x_0, y_0)
Let w = f(x, y), x = g(r, s), y = h(r, s) then:
\frac{\partial w}{\partial r} = \frac{\partial w}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial w}{\partial y} \frac{\partial y}{\partial r}
Suppose that F(x, y) is differentiable and that the equation F(x, y) = 0 defines y implicitly as a differentiable function of z, Then at any point where F_y \ne 0:
\frac{dy}{dx} = - \frac{F_x}{F_y}
Direction of the steepest ascent
\nabla f(x, y) = grad f(x, y) = [f_x(x, y), f_y(x, y)] = f_x(x, y)\mathbf{i} + f_y(x, y)\mathbf{j}
Rate of change along a unit vector: \nabla_{\vec u} f(x_0, y_0) = \nabla f(x_0, y_0) \cdot \vec u = |\nabla f(x_0, y_0)| \cos \theta = \lim_{s \to 0^+} \frac{f(P_0 + s \vec u) - f(P_0)}{s}
Like a tangent line, but it’s a plane because of the higher dimension of the graph. The tangent plane at P_0 = (x_0, y_0, f(x_0, y_0)) = (x_0, y_0, z_0) is defined as:
f_x(P_0)(x - x_0) + f_y(P_0)(y - y_0) - (z - z_0) = 0
Peaks and valleys
minimum/maximum in a given radius
minimum/maximum across the whole domain
If f(a, b) is a local extremum then f_x(a, b) = f_y(a, b) = 0
Points where f_x and f_y are 0 or do not exist are called critical points of f. Extrema of f can only exist on such points or on boundaries. However not every critical point means that theres an extremum there.
When there is a local maximum and minimum at the same time at a critical point.
Df(x, y) = \begin{vmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \\ \end{vmatrix} = f_{xx}f_{yy} - f^2_{xy}
Let f(a, b) be a critical point, then: