Intro to Calculus Flashcards
Given the ODE
dy/dx + p(x)y = q(x)
What is the integrating factor I(x)?
I(x) = e^∫p(x) dx
What is the complementary function to an ODE?
The solution to the homogeneous equation
Second order linear ODEs:
given the auxillary equation
am² + bm + c
What is the solution to the ode when b² - 4ac < 0?
Roots = a +- bi
y = eᵃˣ( Acosbx + Bsinbx )
Second order linear ODEs:
given the auxillary equation
am² + bm + c
What is the solution to the ode when b² - 4ac = 0?
Root = a
y = (A + Bx)eᵃˣ
Second order linear ODEs:
given the auxillary equation
am² + bm + c
What is the solution to the ode when b² - 4ac > 0?
Roots = a, b
y = Aeᵃˣ + Beᵇˣ
What is the form of the particular solution given
ODE = eˣ(sinx)?
yₚ(x) = eˣ(asinx + bcosx)
What is the form of the particular solution given
ODE = sin²x?
yₚ(x) = a + bcos2x + csin2x
∂f/∂xᵢ (p₁, …, pₙ) = ???
∂f/∂xᵢ (p₁, …, pₙ) = lim(h→0) (f(p₁, …, pᵢ₋₁, pᵢ + h, pᵢ₊₁, …, pₙ) - f(p₁, …, pₙ))/h
Given F(x, y) = f(u(x, y), v(x, y)) what is ∂F/∂x?
∂F/∂x = (∂f/∂u)(∂u/∂x) + (∂f/∂v)(∂v/∂x)
Let F(t) = f(u(t), v(t)) with u and v differentiable and f being continuously differentiable in each variable. Then dF/dt = ??
(∂f/∂u)(du/dt) + (∂f/∂v)(dv/dt)
Notice the ds and ∂s
What is the general solution to the PDE:
∂²f/∂y∂x = 0
f(x, y) = P(x) + Q(y),
What is the general solution to the PDE:
∂²f/∂x² = 0
f(x, y) = xp(y) + q(y)
What is the general solution to the PDE:
uₓₓ − u = 0
Since there are no derivatives with respect to y, we can solve the associated ODE
d²u/dx²− u = 0,
where u is treated as being a function of x only. This ODE has solution u(x) = C₁eˣ + C₂e⁻ˣ,
where C₁ and C₂ are constants, and so the solution to the original PDE is
u(x, y) = A(y)eˣ + B(y)e⁻ˣ,
where A and B are arbitrary functions of y only
Find all solutions of the form T(x, t) = A(x)B(t) to the one-dimensional heat/diffusion
equation
∂T/∂t = κ ∂²T/∂x²
where κ is a positive constant, called the thermal diffusivity.
If we substitute
T(x, t) = A(x)B(t)
into the heat equation we find
A(x)B’(t) = κA′′(x)B(t).
Be very careful with the prime here; it denotes the derivative with respect to the independent
variable in question, so that A′(x) denotes dA/dx and B′(t) denotes dB/dt, etc
If we separate the variables we obtain
A′′(x)/A(x) = B′(t)/κB(t) = c, where we assume A ≠ 0 and B ≠ 0
c must be a constant
{ C₁e^(√(c)x) + D₁e^(-√(c)x) c > 0
A(x) = { C₂x + D₂ c = 0
{ C₃cos(√(-c)x) + D₃sin(√(-c)x) c < 0
What is the Jacobian ∂(x, y)/∂(u, v)?
|∂x/∂u ∂x/∂v|
|∂y/∂u ∂y/∂v|
If A is a domain in the xy-plane mapped one to one and onto a domain B in the uv-plane then
∫ₐ f(x, y) dx dy = ???
∫ₐ f(x, y) dx dy = ∫ᵦ f(x(u, v), y(u, v)) |∂(x, y)/∂(u, v)| du dv
( uₓ uᵧ)⁻¹ = ??
vₓ vᵧ
( uₓ uᵧ)⁻¹ = ( xᵤ xᵥ )
vₓ vᵧ) ( yᵤ yᵥ
Let R ⊂ R². Then we define the area of R to be
A(R) =
A(R) = ∫∫₍ₓ,ᵧ₎∈ᵣ dx dy
Let f : R → S be a bijection between two regions of R², and write (u, v) = f(x, y).
A(S) = ∫∫₍ᵤ,ᵥ₎∈ₛ du dv = ???
A(R) = ∫∫₍ₓ,ᵧ₎∈ᵣ dx dy = ??
A(S) = ∫∫₍ᵤ,ᵥ₎∈ₛ du dv = ∫∫₍ₓ,ᵧ₎∈ᵣ |∂(u, v)/∂(x, y)| dx dy
A(R) = ∫∫₍ₓ,ᵧ₎∈ᵣ dx dy = ∫∫₍ᵤ,ᵥ₎∈ₛ |∂(x, y)/∂(u, v)| du dv
What is the equation of a Circle?
What is the parametrisation?
Area?
x² + y² = a²
parametrisation (x, y) = (a cos θ, a sin θ); area πa²
What is the equation of an Ellipse? What is the parametrisation? Eccentricity? Foci? Directrices? Area?
x²/a² + y²/b² = 1 (b < a) parametrisation (x, y) = (a cos θ, b sin θ); eccentricity e =√ (1 − b²/a²); foci (±ae, 0); directrices x = ±a/e; area πab
What is the equation of a Parabola? What is the parametrisation? Eccentricity? Foci? Directrices?
y² = 4ax
parametrisation (x, y) = (at², 2at); eccentricity e = 1; focus (a, 0); directrix x = −a
What is the equation of a Hyperbola? What is the parametrisation? Eccentricity? Foci? Directrices? Asymptotes?
x²/a² - y²/b² = 1
parametrisation (x, y) = (a sec t, b tan t); eccentricity e =√ (1 + b²/a²);
foci (±ae, 0);
directrices x = ±a/e; asymptotes y = ±bx/a.
What is the equation of a Sphere?
x² + y² + z² = a²
What is the equation of an Ellipsoid?
x²/a² + y²/b² + z²/c² = 1
What is the equation of a Hyperboloid of one sheet?
x²/a² + y²/b² - z²/c² = 1
What is the equation of a Hyperboloid of two sheets?
x²/a² - y²/b² - z²/c² = 1
What is the equation of a Paraboloid?
x² + y² = z
What is the equation of a Hyperbolic paraboloid?
x² - y² = z
What is the equation of a Cone?
x² + y² = z²
What is a smooth parametrised surface?
A smooth parametrised surface is a map r, given by the parametrisation
r : U → R³: (u, v) |→ (x(u, v), y(u, v), z(u, v)),
from an open subset U ⊆ R² to R³ such that
• x, y, z have continuous partial derivatives with respect to u and v of all orders;
• r is a bijection, with both r and r⁻¹ being continuous;
• at each point the vectors
∂r/∂u and ∂r/∂v
are linearly independent
What is a tangent plane?
Let r : U → R³ be a smooth parametrised surface and let p be a point on the surface.
The plane containing p and which is parallel to the vectors ∂r/∂u(p) and ∂r/∂v (p) is called the tangent plane to r(U) at p
Let r : U → R³ be a smooth parametrised surface and let p be a point on the surface
Give an equation of a normla to the surface
∂r/∂u(p) ∧ ∂r/∂v (p)
What is a scalar line integral?
The scalar line integral of a vector field F(r) along a path C given by r = r(t), from r(t₀) to r(t₁), is ∫𝒸 F(r) · dr = ₜ₀∫ᵗ¹ F(t) · dr/dt dt
How do you calculate the length of a curve?
∫𝒸 F(r) · dr = ₜ₀∫ᵗ¹ |r˙(t)| dt = ₜ₀∫ᵗ¹√ ((dx/dt)² + (dy/dt)² ) dt
Given a scalar function f := Rⁿ → R, whose partial derivatives all exist, the
gradient vector ∇f = ??
∇f = (∂f/∂x₁, …, ∂f/∂xₙ)
ₐ∫ᴮ ∇f· dr = ???
ₐ∫ᴮ ∇f· dr = f(B) − f(A)
Let f : Rⁿ → R be a differentiable scalar function and let u be a unit vector. Then the directional derivative of f at a in the direction u equals??
lim(t→0) ( (f(a + tu) − f(a))/t)
This is the rate of change of the function f at a in the direction u.
The directional derivative of a function f at the point a in the direction u equals??
In terms of grad f
∇f(a) · u.
Let f : Rⁿ → R be a differentiable scalar function and let u be a unit vector.
The rate of change of f is greatest in the direction ∇f, that is when u = ??, and the maximum rate of change is given by [ ]
The rate of change of f is greatest in the direction ∇f, that is when u = ∇f /|∇f|, and the maximum rate of change is given by |∇f|
What is a level set of a function f : R³ → R ?
is a set of points
{(x, y, z) ∈ R³: f(x, y, z) = c},
where c is a constant. For suitably well behaved functions f and constants c, the level set is a
surface in R³
Given a surface S ⊆ R³ with equation f(x, y, z) = c and a point p ∈ S
What is the equation for the normal to S at p?
∇f(p)
Let f and g be differentiable functions of x, y, z. Then
- ∇(fg) = [ ]
- ∇(fⁿ) = [ ]
- ∇(f/g) = [ ]
- ∇(f(g(x))) = [ ]
- ∇(fg) = f∇g + g∇f;
- ∇(fⁿ) = nfⁿ⁻¹∇f;
- ∇(f/g) = (g∇f − f∇g)/g²;
- ∇(f(g(x))) = f’(g(x))∇g(x)
What is Taylor’s theorem for a function of one variable?
f(x) = f(a) + f’(a)(x − a) + · · · +
f⁽ⁿ⁾(a)(x − a)ⁿ/n! + f⁽ⁿ⁺¹⁾(ξ)(x − a)ⁿ⁺¹/(n + 1)!
The first-order Taylor expansion for f(x, y) about (a, b) is ….
p₁(x, y) = f(a, b) + fₓ(a, b)(x − a) + fᵧ(a, b)(y − b).
The second-order Taylor expansion for f(x, y) about (a, b) is …
p₂(x, y) = f(a, b) + fₓ(a, b)(x − a) + fᵧ(a, b)(y − b) + 1/2 [fₓₓ(a, b)(x − a)² + 2fₓᵧ(a, b)(x − a)(y − b) + fᵧᵧ(a, b)(y − b)²]
What is the general formula for Taylor’s theorem for a function of two variables?
pg 59
Suppose that f(x, y) defined on an open subset U has continuous partial derivatives, and (x₀, y₀) ∈ U is a local maximum or a local minimum. Then
∂f/∂x(x₀, y₀) = [ ]
∇f(x₀, y₀) = ?
∂f/∂x(x₀, y₀) = ∂f/∂y (x₀, y₀) = 0
∇f(x₀, y₀) = 0
Suppose that f(x, y) is defined on an open subset U and has continuous derivatives
up to second order, and suppose that (x₀, y₀) ∈ U is a critical point, i.e. ∇f(x₀, y₀) = 0
Define a local maximum and minimum
1) If fₓₓ(x₀, y₀)fᵧᵧ(x₀, y₀) - (fₓᵧ(x₀, y₀))² > 0, fₓₓ(x₀, y₀) < 0
then (x₀, y₀) is a local maximum.
2) If
fₓₓ(x₀, y₀)fᵧᵧ(x₀, y₀) - (fₓᵧ(x₀, y₀))² > 0, fₓₓ(x₀, y₀) > 0
then (x₀, y₀) is a local minimum.
Stationary points:
What if fₓₓfᵧᵧ - f²ₓᵧ < 0?
Saddle
Suppose f(x) is a function with n variables x = (x₁, . . . , xₙ) defined on an open subset U ⊂ Rⁿ which has continuous partial derivatives up to second order. Let a = (a₁, . . . , aₙ) be a critical point, that is, ∇f(a) = 0 What does it mean if the Hessian matrix H(a) is positive definite?
1) If the Hessian matrix H(a) is positive definite, then a is a local minimum of f
Suppose f(x) is a function with n variables x = (x₁, . . . , xₙ) defined on an open subset U ⊂ Rⁿ which has continuous partial derivatives up to second order. Let a = (a₁, . . . , aₙ) be a critical point, that is, ∇f(a) = 0 What does it mean if the Hessian matrix H(a) is negitive definite?
2) If the Hessian matrix H(a) is negative definite, then a is a local maximum of f
What is a Hessian matrix?
H = | f²ₓ₁ₓ₁ f²ₓ₁ₓ₂ …. f²ₓ₁ₓₙ |
| f²ₓ₂ₓ₁ |
| …. …. |
….
| f²ₓₙₓ₁ f²ₓₙₓₙ |
How do you find the critical points of f(x, y, z) given it’s constrained by F(x, y, z)
∇f(x, y, z) = λ∇F(x, y, z)
We introduce a function G(x, y, z, λ) = f(x, y, z) − λF(x, y, z). Then the system may be written as
∂G/∂x = ∂G/∂y = ∂G/∂z = ∂G/∂λ = 0
