Added portion of multivariable calculus.

config/en/mkdocs.yaml

@@ -54,12 +54,21 @@ nav:
       - 'Integration': mathematicse/calculus/integration.md
       - 'Integration techniques': mathematics/calculus/integration-techniques.md
       - 'Improper integrals': mathematics/calculus/improper-integrals.md
+    - 'Multivariable calculus':
+      - 'Functions of several variables': mathematics/multivariable-calculus/functions-of-several-variables.md
+      - 'Limits and continuity': mathematics/multivariable-calculus/limits-and-continuity.md
+      - 'Differentation': mathematics/multivariable-calculus/differentation.md
+      - 'Implicit equations': mathematics/multivariable-calculus/implicit-equations.md
+      - 'Taylor polynomials': mathematics/multivariable-calculus/taylor-polynomials.md
+      - 'Extrema': mathematics/multivariable-calculus/extrema.md
+      - 'Integration': mathematics/multivariable-calculus/integration.md
     - 'Ordinary differential equations':
       - 'First order differential equations': mathematics/ordinary-differential-equations/first-order-ode.md
       - 'Second order differential equations': mathematics/ordinary-differential-equations/second-order-ode.md
       - 'Systems of linear differential equations': mathematics/ordinary-differential-equations/systems-of-linear-ode.md
       - 'The Laplace transform': mathematics/ordinary-differential-equations/laplace-transform.md
     
+
   - 'Physics':
     - 'Start': physics/start.md
     

docs/en/mathematics/multivariable-calculus/differentation.md

@@ -0,0 +1,127 @@
+# Differentation
+
+Generalization of derivatives to higher dimensions:
+
+* limit of difference quotient: partial derivatives,
+* linearization: total derivative.
+
+## Partial derivatives
+
+*Definition*: let $D \subseteq \mathbb{R}^n$ ($n=2$ for simplicity) and let $f: D \to \mathbb{R}$ and $\mathbf{a} \in D$, if the limit exists the partial derivates of $f$ are
+
+$$
+\begin{align*}
+    &\partial_1 f(\mathbf{a}) := \lim_{h \to 0} \frac{f(a_1 + h, a_2) - f(\mathbf{a})}{h}, \\
+    &\partial_2 f(\mathbf{a}) := \lim_{h \to 0} \frac{f(a_1, a_2 + h) - f(\mathbf{a})}{h}.
+\end{align*}
+$$
+
+*Theorem*: suppose that two mixed $n$th order partial derivatives of a function $f$ involve the same differentations but in different orders. If those partials are continuous at a point $\mathbf{a}$ and if $f$ and all partials of $f$ of order less than $n$ are continuous in a neighbourhood of $\mathbf{a}$, then the two mixed partials are equal at the point $\mathbf{a}$. We have for $n=2$ 
+
+$$
+    \partial_{12} f(P) = \partial_{21} f(P),
+$$
+
+*Proof*: will be added later.
+
+## Total derivatives
+
+*Definition*: let $D \subseteq \mathbb{R}^n$ ($n=2$ for simplicity) and let $f: D \to \mathbb{R}$, determining an affine linear approximation of $f$ around $\mathbf{a} \in D$
+
+$$
+    p(\mathbf{x}) = f(\mathbf{a}) + \big\langle L,\; \mathbf{x} - \mathbf{a} \big\rangle,
+$$
+
+with $f(\mathbf{x}) = p(\mathbf{x}) + r(\mathbf{x})$ demand $\frac{r(\mathbf{x})}{\|\mathbf{x} - \mathbf{a}\|} \to 0$ when $\mathbf{x} \to \mathbf{a}$. 
+
+if $L \in \mathbb{R}^2$ exists to satisfy this, then $f$ is called totally differentiable in $\mathbf{a}$. 
+
+*Theorem*: if $f$ is totally differentiable in $\mathbf{a}$, then $f$ is partially differentiable in $\mathbf{a}$ and the partial derivatives are
+
+$$
+    \partial_1 f(\mathbf{x}) = L_1, \qquad \partial_2 f(\mathbf{x}) = L_2,
+$$
+
+obtaining 
+
+$$
+    p(\mathbf{x}) = f(\mathbf{a}) + \big\langle \nabla f(\mathbf{a}),\; \mathbf{x} - \mathbf{a} \big\rangle.
+$$
+
+with $\nabla f(\mathbf{a})$ the gradient of $f$.
+
+## Gradients
+
+*Definition*: at any point $\mathbf{x} \in D$ where the first partial derivatives of $f$ exist, we define the gradient vector $\nabla$ by
+
+$$
+    \nabla f(\mathbf{x}) = \begin{pmatrix} \partial_1 f(\mathbf{x}) \\ \partial_2 f(\mathbf{x}) \end{pmatrix}.
+$$
+
+The direction of the gradient is the direction of steepest increase of $f$ at $\mathbf{x}$.
+
+<br>
+
+*Theorem*: gradients are orthogonal to level lines and level surfaces.
+
+*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
+
+$$
+    \partial_1 f(\mathbf{x}) \dot x(t) + \partial_2 f(\mathbf{x}) \dot y(t) = 0,
+$$
+
+at $t=0$, we can rewrite this to
+
+$$
+    \big\langle \nabla f(\mathbf{a}),\; \mathbf{\dot r}(0) \big\rangle = 0,
+$$
+
+obtaining that $\nabla f$ is orthogonal to $\mathbf{\dot r}$. 
+
+## Directional derivatives
+
+*Definition*: let $D \subseteq \mathbb{R}^n$ and let $f: D \to \mathbb{R}$ with $\mathbf{v} \in D$ and $\|\mathbf{v}\| = 1$ a unit vector. The directional derivative is then the change of $f$ near a point $\mathbf{a} \in D$ in the direction of $\mathbf{v}$ 
+
+$$  
+    D_\mathbf{v} f(\mathbf{a}) = \big\langle \mathbf{v},\; \nabla f(\mathbf{a}) \big\rangle.
+$$
+
+## The general case
+
+*Definition*: let $D \subseteq \mathbb{R}^n$ and let $\mathbf{f}: D \to \mathbb{R}^m$, with $f_i: D \to \mathbb{R}$, with $i = 1, \dotsc, m$ being the components of $\mathbf{f}$. 
+
+* $\mathbf{f}$ is continuous at $\mathbf{a} \in D$ $\iff$ all $f_i$ continuous at $\mathbf{a}$,
+* $\mathbf{f}$ is partially/totally differentiable at $\mathbf{a}$ $\iff$ all $f_i$ are partially/totally differentiable at $\mathbf{a}$. 
+
+The linearization of every component $f_i$ we have
+
+$$
+    f_i(\mathbf{x}) = f_i(\mathbf{a}) + \big\langle \nabla f_i(\mathbf{a}),\; \mathbf{x} - \mathbf{a} \big\rangle + r_i(\mathbf{x}),
+$$
+
+so in total we have
+
+$$
+    \mathbf{f}(\mathbf{x}) = \mathbf{f}(\mathbf{a}) + D\mathbf{f}(\mathbf{a}) \big(\mathbf{x} - \mathbf{a}\big) + \mathbf{r}(\mathbf{x}),
+$$
+
+with $D\mathbf{f}(\mathbf{a})$ the Jacobian of $\mathbf{f}$.
+
+*Definition*: the Jacobian is given by $\big[D\mathbf{f}(\mathbf{a}) \big]_{i,\;j} = \partial_j f_i(\mathbf{a}).$
+
+### Chain rule
+
+Let $D \subseteq \mathbb{R}^n$ and let $E \subseteq \mathbb{R}^m$ be sets and let $\mathbf{f}: D \to \mathbb{R}^m$ and let $\mathbf{g}: E \to \mathbb{R}^k$ with $\mathbf{f}$ differentiable at $\mathbf{x}$ and $\mathbf{g}$ differentiable at $\mathbf{f}(\mathbf{x})$. Then $D\mathbf{f}(\mathbf{x}) \in \mathbb{R}^{m \times n}$ and $D\mathbf{g}\big(\mathbf{f}(\mathbf{x})\big) \in \mathbb{R}^{k \times m}$. 
+
+Then if we differentiate $\mathbf{g} \circ \mathbf{f}$ we obtain
+
+$$
+    D(\mathbf{g} \circ \mathbf{f})(\mathbf{x}) = D\mathbf{g}\big(\mathbf{f}(\mathbf{x})\big) D\mathbf{f}(\mathbf{x}).
+$$
+
+We have two interpretations:
+
+* the composition of linear maps,
+* the matrix multiplication of the Jacobian.
+
+*Proof*: will be added later.

docs/en/mathematics/multivariable-calculus/functions-of-several-variables.md

@@ -0,0 +1,21 @@
+# Functions of several variables
+
+*Definition*: let $D \subseteq \mathbb{R}^m$ with $m>1$, and $f: D \to \mathbb{R}^n$ then $f$ is a function of several variables where:
+
+* for $n=1$, $f$ is a scalar function,
+* for $n>1$, $f$ is a vector valued function.
+
+<br>
+
+*Definition*: the domain convention specifies that the domain of a function of $m$ variables is the largest set of points for which the function makes sense as a real number, unless that domain is explicitly stated to be a smaller set. 
+
+## Graphical representations of scalar valued functions
+
+### Graphs
+
+*Definition*: let $D \subseteq \mathbb{R}^2$ and let $f: D \to \mathbb{R}$ then $G_f := \big\{\big(x, y, f(x,y)\big) \;\big|\; (x, y) \in D\big\}$ is the graph of $f$. Observe that $G_f \subseteq \mathbb{R}^3$. 
+
+### Level sets
+
+*Definition*: let $D \subseteq \mathbb{R}^2$ and let $f: D \to \mathbb{R}$ then for $c \in \mathbb{R}$ we have $S_c := \big\{(x, y) \in D \;\big|\; f(x,y) = c \big\}$ is the level set of $f$. Observe that $S_c \subseteq \mathbb{R}^2$. 
+

docs/en/mathematics/multivariable-calculus/limits-and-continuity.md

@@ -0,0 +1,21 @@
+# Limits and continuity
+
+## Limit
+
+*Definition*: let $D \subseteq \mathbb{R}^m$ and let $f: D \to \mathbb{R}^n$, with $m,n \in \mathbb{N}$. Let $\mathbf{a}$ be the point $\mathbf{x}$ approaches, then $f$ approaches the limit $L \in \mathbb{R}^n$
+
+$$
+\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = L \iff \forall \varepsilon_{>0} \exists \delta_{>0} \Big[0 < \|\mathbf{x} - \mathbf{a}\|< \delta \implies \|f(\mathbf{x}) - L\| < \varepsilon \Big],
+$$
+
+with $\mathbf{a}, \mathbf{x} \in \mathbb{R}^m$.
+
+## Continuity
+
+*Definition*: let $D \subseteq \mathbb{R}^m$ and let $f: D \to \mathbb{R}^n$, with $m,n \in \mathbb{N}$. Then $f$ is called continuous at $\mathbf{a}$ if
+
+$$
+\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = f(\mathbf{a}),
+$$
+
+with $\mathbf{a}, \mathbf{x} \in \mathbb{R}^m$.

docs/en/mathematics/ordinary-differential-equations/laplace-transform.md

@@ -141,7 +141,7 @@ $$
 \begin{align*}
     &Y_1(s) = \frac{\mathrm{det}\begin{pmatrix} 3 & 1 \\ -7 & 5 - s \end{pmatrix}}{(5-s^2)-1}, \\
     \\
-    &Y_2(s) = \frac{\mathrm{det}\begin{pmatrix} 5 - s & 3 \\ 1 & -7\end{pmatrix}}{(5-s^2)-1}
+    &Y_2(s) = \frac{\mathrm{det}\begin{pmatrix} 5 - s & 3 \\ 1 & -7\end{pmatrix}}{(5-s^2)-1},
 \end{align*}
 $$