Calculus of several variables Flashcards
Domain
The set D = D(f) is the set of values x can take in the function f(x)
Range
The set R = R(f) := {f(x) : x ∈ D(f )} is the set of values the function can output.
Maximal domain
The broadest domain
euclidian length
for x = (x,y) |x|= √x² +y²
euclidian distance
for x₁ = (x₁,y₁) and x₂ = (x₂,y₂ )
we have |x₁ - x₂|= √((x₁ -x₂)² + (y₁ - y₂)²)
∂-neighbourhood
The ∂-neighbourhood of a point x0 is a set of all points of x such that |x-x0| < ∂ Denoted U∂(x0)
Interior point of D
A point x0 is called an interior point of set D if there exists a ∂>0 such that U∂(x0) ⊂ D
exterior point of D
A point x0 is called an exterior point of set D if there exists a ∂ >0 such that U∂(x0) ⊂ R ͩ \ D
Boundary point of D
A point x0 is called a boundary point of set D if for any ∂> 0 U∂(x0) contains both points of D and R ͩ \ D
Open set
A set D is called open if all its points are interior
Closed set
A set D is closed if it contains all its boundary points
limit of a function
we can say l is a limit of a function f at a point x0 if for any ɛ>0 there exists a ∂ > 0 such that for any x in U∂(x0) one has |f(x) - l |< ɛ
limx->x0 f(x) = L
when is a function continuous
a function f is continuous at a point x0 if
limx->x0 f(x) = f(x0)
f is continuous on a set S if f is continuous at every point of set S.
How do you prove f(x) has a limit
-Use the limit definition
-By algebra of limits
-squeeze theorem
partial derivatives
the partial derivative is obtained by diff f with respect to x treating y as a constant.
fₓ =(x,y) limh->0 (f(x+h,y)-f(x,y))/h fy (x,y) = limh->0 (f(x,y+h)-f(x,y))/h
Higher order partial derivatives
fₓₓ = ∂² f/∂x² = ∂/∂x (∂f/∂x), you can change the variable we are differentiating with respect to
little-o(h)
A function is said to be a little-o(h) or simply o(h) if limh->0 g(h)/|h|=0
Function increment (little-o(h))
The function increment or the change in value of a function can be written as f(x+h) - f(x) = A x h + o(h) where A is a constant vector which is known as the gradient of f at x, denoted ▽f(x)
Directional derrivative
for each unit vector u where |u|=1 the limit
fu’(x) = limt->0 (f(x+tu) - f(x))/t
if it exists it is the directional derivative at point x in the direction u
derivatives (partial, directional)
The partial derivative gives the rate of change along the corresponding coordinate
The directional derivative gives the rate of change in the direction u at point x
If function is differentiable at x the f has d derivatives in every direction u and fu’(x) = ▽f(x) x u
chain rule
d/dt (f(r(t))) = ▽f(r(t)) x r’(t)
or
df/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) + (∂f/∂z)(dz/dt)
where r(t) = (x(t),y(t),z(t))
