diff --git a/docs/en/mathematics/multivariable-calculus/differentation.md b/docs/en/mathematics/multivariable-calculus/differentation.md
index 67650bc..6521dff 100644
--- a/docs/en/mathematics/multivariable-calculus/differentation.md
+++ b/docs/en/mathematics/multivariable-calculus/differentation.md
@@ -22,12 +22,9 @@ $$
\partial_{12} f(P) = \partial_{21} f(P),
$$
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
## Total derivatives
@@ -55,12 +52,9 @@ $$
with $\nabla f(\mathbf{a})$ the gradient of $f$.
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
## Chain rule
@@ -90,24 +84,21 @@ The direction of the gradient is the direction of steepest increase of $f$ at $\
*Theorem*: gradients are orthogonal to level lines and level surfaces.
-
-Proof:
+??? note "*Proof*:"
-let $\mathbf{r}(t) = \big(x(t),\; y(t) \big)^T$ be a parameterization of the level curve of $f$ such that $\mathbf{r}(0) = \mathbf{a}$. Then for all $t$ near $0$, $f(\mathbf{r}(t)) = f(\mathbf{a})$. Differentiating this equation with respect to $t$ using the chain rule, we obtain
+ let $\mathbf{r}(t) = \big(x(t),\; y(t) \big)^T$ be a parameterization of the level curve of $f$ such that $\mathbf{r}(0) = \mathbf{a}$. Then for all $t$ near $0$, $f(\mathbf{r}(t)) = f(\mathbf{a})$. Differentiating this equation with respect to $t$ using the chain rule, we obtain
-$$
- \partial_1 f(\mathbf{x}) \dot x(t) + \partial_2 f(\mathbf{x}) \dot y(t) = 0,
-$$
+ $$
+ \partial_1 f(\mathbf{x}) \dot x(t) + \partial_2 f(\mathbf{x}) \dot y(t) = 0,
+ $$
-at $t=0$, we can rewrite this to
+ at $t=0$, we can rewrite this to
-$$
- \big\langle \nabla f(\mathbf{a}),\; \mathbf{\dot r}(0) \big\rangle = 0,
-$$
+ $$
+ \big\langle \nabla f(\mathbf{a}),\; \mathbf{\dot r}(0) \big\rangle = 0,
+ $$
-obtaining that $\nabla f$ is orthogonal to $\mathbf{\dot r}$.
-
-
+ obtaining that $\nabla f$ is orthogonal to $\mathbf{\dot r}$.
## Directional derivatives
@@ -155,10 +146,6 @@ We have two interpretations:
* the composition of linear maps,
* the matrix multiplication of the Jacobian.
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
-
\ No newline at end of file
+ Will be added later.
\ No newline at end of file
diff --git a/docs/en/mathematics/multivariable-calculus/extrema.md b/docs/en/mathematics/multivariable-calculus/extrema.md
index 1ab679e..58ac181 100644
--- a/docs/en/mathematics/multivariable-calculus/extrema.md
+++ b/docs/en/mathematics/multivariable-calculus/extrema.md
@@ -8,12 +8,9 @@
*Theorem*: if $f$ has local $\begin{matrix} \text{ maximum } \\ \text{ minimum } \end{matrix}$ at $\mathbf{x^*} \in D$ then $\mathbf{x^*}$ is a critical point of for $f$.
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
## A second derivative test
@@ -30,23 +27,17 @@ $$
* If $H_f(\mathbf{x^*})$ is indefinite (both positive and negative eigenvalues), then $f$ has a saddle point at $\mathbf{x^*}$.
* If $H_f(\mathbf{x^*})$ is neither positive nor negative definite, nor indefinite, (eigenvalues equal to zero) this test gives no information.
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
## Extrema on restricted domains
*Theorem*: let $D \subseteq \mathbb{R}^n$ be bounded and closed ($D$ contains all boundary points). Let $f: D \to \mathbb{R}$ be continuous, then $f$ has a global maximum and minimum.
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
**Procedure to find the global maximum and minimum**:
@@ -68,12 +59,9 @@ $$
L(\mathbf{x}, \lambda) := f(\mathbf{x}) - \lambda g(\mathbf{x}).
$$
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
### The general case
@@ -89,12 +77,9 @@ $$
L(\mathbf{x}, \mathbf{\lambda}) := f(\mathbf{x}) - \big\langle \mathbf{\lambda},\; \mathbf{g}(\mathbf{x}) \big\rangle.
$$
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
#### Example
diff --git a/docs/en/mathematics/multivariable-calculus/implicit-equations.md b/docs/en/mathematics/multivariable-calculus/implicit-equations.md
index bb71128..b278e5f 100644
--- a/docs/en/mathematics/multivariable-calculus/implicit-equations.md
+++ b/docs/en/mathematics/multivariable-calculus/implicit-equations.md
@@ -26,12 +26,9 @@ $$
\phi' (x) = - \frac{\partial_1 f\big(x,\phi(x)\big)}{\partial_2 f\big(x,\phi(x)\big)}.
$$
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
## General case
@@ -56,9 +53,6 @@ $$
D \mathbf{\phi}(\mathbf{x}) = - \Big(D_2 \mathbf{F}\big(\mathbf{x},\mathbf{\phi}(\mathbf{x})\big) \Big)^{-1} D_1 \mathbf{F}\big(\mathbf{x},\mathbf{\phi}(\mathbf{x})\big).
$$
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
diff --git a/docs/en/mathematics/multivariable-calculus/integration.md b/docs/en/mathematics/multivariable-calculus/integration.md
index db39c71..d78204e 100644
--- a/docs/en/mathematics/multivariable-calculus/integration.md
+++ b/docs/en/mathematics/multivariable-calculus/integration.md
@@ -8,12 +8,9 @@ $$
implying that order can be interchanged, this is true for $n \in \mathbb{N}$.
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
## Iteration of integrals
@@ -23,12 +20,9 @@ $$
\iint_D f dA = \iint_R F dA, \qquad \text{where } F(\mathbf{x}) = \begin{cases} F(\mathbf{x}) \quad &\mathbf{x} \in D, \\ 0 \quad &\mathbf{x} \notin D. \end{cases}
$$
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
## Coordinate transformation for integrals
@@ -46,12 +40,9 @@ $$
with $D_\phi$ the Jacobian of $\phi$.
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
### Example
diff --git a/docs/en/mathematics/multivariable-calculus/taylor-polynomials.md b/docs/en/mathematics/multivariable-calculus/taylor-polynomials.md
index 0b668cf..018b17a 100644
--- a/docs/en/mathematics/multivariable-calculus/taylor-polynomials.md
+++ b/docs/en/mathematics/multivariable-calculus/taylor-polynomials.md
@@ -24,17 +24,13 @@ $$
f(\mathbf{x}) = T(\mathbf{x}) + \frac{1}{(n+1)!} \partial_\mathbf{h}^{n+1} f(\mathbf{a} + \theta \mathbf{h}).
$$
-
-Proof:
+??? note "*Proof*:"
-apply Taylor’s theorem in 1D and the chain rule to the function $\phi : [0, 1] \to \mathbb{R}$ given by
+ Apply Taylor’s theorem in 1D and the chain rule to the function $\phi : [0, 1] \to \mathbb{R}$ given by
-$$
- \phi(\theta) := f(\mathbf{a} + \theta \mathbf{h}).
-$$
-
-
-
+ $$
+ \phi(\theta) := f(\mathbf{a} + \theta \mathbf{h}).
+ $$
## Other methods
diff --git a/docs/en/mathematics/ordinary-differential-equations/laplace-transform.md b/docs/en/mathematics/ordinary-differential-equations/laplace-transform.md
index c40ead1..d671a94 100644
--- a/docs/en/mathematics/ordinary-differential-equations/laplace-transform.md
+++ b/docs/en/mathematics/ordinary-differential-equations/laplace-transform.md
@@ -18,12 +18,9 @@ $$
on the interval where both are defined.
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
If $c \in \mathbb{R}$ then $cf$ also has a Laplace transform and,
@@ -45,12 +42,9 @@ $$
on this interval
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
**More shifting**: let $a>0$, if $f$ has a Laplace transform $F$ on $s_0, \infty$ then the function $g$ given by
@@ -66,12 +60,9 @@ $$
on this interval.
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
**Scaling**: let $a > 0$. If $f$ has a Laplace transform $F$ on $(s_0, \infty)$ then the function $g$ given by
@@ -87,12 +78,9 @@ $$
on this interval.
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
**Derivatives**: if $f$ has a derivative $g$ having a Laplace transform $G$ on the interval $(s_0,\infty)$ then $f$ has a Laplace transform on the same interval, and
@@ -106,23 +94,19 @@ $$
\mathcal{L}[f^{(n)}](s) = s^n F(s) - \sum_{k=0}^{n-1} s^k f^{(n-1-k)}(0)
$$
-
-Proof:
+??? note "*Proof*:"
-for large enough $s$, the case $n=1$ follows by integration by parts
+ For large enough $s$, the case $n=1$ follows by integration by parts
-$$
-\begin{align*} \mathcal{L}[f'](s) &= \int_0^\infty e^{-st} f'(t)dt, \\ &= \Big[e^{-st} f(t) \Big]_0^\infty + s\int_0^\infty e^{-st}f(t), \\ &= sF(s) - f(0) \end{align*},
-$$
+ $$
+ \begin{align*} \mathcal{L}[f'](s) &= \int_0^\infty e^{-st} f'(t)dt, \\ &= \Big[e^{-st} f(t) \Big]_0^\infty + s\int_0^\infty e^{-st}f(t), \\ &= sF(s) - f(0) \end{align*},
+ $$
-suppose $\mathcal{L}[f^{k}](s) = s^k F(s) - \sum_{r=0}^{k-1} s^r f^{(k-1-r)}(0)$ is true for $k \in \mathbb{N}$, then by assumption
+ suppose $\mathcal{L}[f^{k}](s) = s^k F(s) - \sum_{r=0}^{k-1} s^r f^{(k-1-r)}(0)$ is true for $k \in \mathbb{N}$, then by assumption
-$$
-\begin{align*} \mathcal{L}[f^{k+1}](s) &= \int_0^\infty e^{-st} f^{(k+1)}(t)dt , \\ &= \Big[e^{-st} f^{(k+1)}(t) \Big]_0^\infty + s\int_0^\infty e^{-st}f^{(k)}(t), \\ &= s \mathcal{L}[f^{(k)}] - f^{(k)}(0), \\ &= s \Big(s^k F(s) - \sum_{r=0}^{k-1} s^r f^{(k-1-r)}(0)\Big) - f^{(k)}(0), \\ &= s^{k+1} F(s) - \sum_{r=0}^{k} s^r f^{(k-r)}(0) \end{align*}.
-$$
-
-
-
+ $$
+ \begin{align*} \mathcal{L}[f^{k+1}](s) &= \int_0^\infty e^{-st} f^{(k+1)}(t)dt , \\ &= \Big[e^{-st} f^{(k+1)}(t) \Big]_0^\infty + s\int_0^\infty e^{-st}f^{(k)}(t), \\ &= s \mathcal{L}[f^{(k)}] - f^{(k)}(0), \\ &= s \Big(s^k F(s) - \sum_{r=0}^{k-1} s^r f^{(k-1-r)}(0)\Big) - f^{(k)}(0), \\ &= s^{k+1} F(s) - \sum_{r=0}^{k} s^r f^{(k-r)}(0) \end{align*}.
+ $$
## Examples
diff --git a/docs/en/mathematics/ordinary-differential-equations/second-order-ode.md b/docs/en/mathematics/ordinary-differential-equations/second-order-ode.md
index 9470db2..9c4645a 100644
--- a/docs/en/mathematics/ordinary-differential-equations/second-order-ode.md
+++ b/docs/en/mathematics/ordinary-differential-equations/second-order-ode.md
@@ -24,8 +24,6 @@ Then the consequence is that the general solution is a linear space.
$(*)$ is said to have **resonance** if $f$ can be split into linearly independent terms of which at least one lies in the solution space of $(*)$.
-
-
### Solving homogeneous linear second-order ODEs with constant coefficients
Therefore solving
@@ -52,12 +50,9 @@ $$
y(t) = (c_1 + c_2t) e^{\lambda_1 t}.
$$
-
-Proof:
+??? note "*Proof*:"
-will be added later.
-
-
+ Will be added later.
#### Example
@@ -73,8 +68,6 @@ $$
y(t) = e^{-2t}\big(d_1\cos 2t + d_2 \sin 2t \big), \quad d_1,d_2 \in \mathbb{R}.
$$
-
-
### Solving inhomogeneous linear second-order ODEs with constant coefficients
*Theorem*: let $y_p$ be a particular solution to $(*)$. Then the general solution to $(*)$ is given by
@@ -85,13 +78,9 @@ $$
with $y_h$ the solution to the homegeneous case.
-
-Proof:
+??? note "*Proof*:"
-let $y$ be a solution to $(*)$, then $L[y - y_p] = L[y] - L[y_p] = f - f = 0$. Therefore $y = (y - y_p) + y_p = y_h + y_p$.
-
-
-
+ Let $y$ be a solution to $(*)$, then $L[y - y_p] = L[y] - L[y_p] = f - f = 0$. Therefore $y = (y - y_p) + y_p = y_h + y_p$.
#### Method of variation of parameters
diff --git a/docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md b/docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md
index b678e44..8deea84 100644
--- a/docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md
+++ b/docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md
@@ -50,13 +50,9 @@ $$
\mathbf{y}(t) = \mathbf{y}_p(t) + \mathbf{y}_h(t), \qquad t \in I
$$
-
-Proof:
+??? note "*Proof*:"
-similar to 1d case, will be added later.
-
-
-
+ Similar to 1d case, will be added later.
### Method of variation of parameters