Minor corrections

vault/Mathematics/Analysis/Real Analysis/Multivariate Real Analysis/Scalar Fields/Differentiation/Gradient.md

@@ -1,9 +1,9 @@
 
 >[!THEOREM] Theorem: Gradient
 >
->Let $f: \mathcal{D} \subseteq \mathbb{R}^n \to \mathbb{R}$ be a [real scalar field](../Real%20Scalar%20Field.md), let $\hat{\mathbf{r}}$ be some [unit vector](../../../../../Algebra/Linear%20Algebra/Vector%20Spaces/Normed%20Vector%20Spaces/Unit%20Vector.md) and let $\mathbf{a} \in \mathcal{D}$.
+>Let $f: \mathcal{D} \subseteq \mathbb{R}^n \to \mathbb{R}$ be a [real scalar field](../Real%20Scalar%20Field.md) and let $\mathbf{a} \in \mathcal{D}$.
 >
->If $f$'s [directional derivative](Directional%20Derivatives%20of%20Real%20Scalar%20Fields.md) $\partial_{\hat{\mathbf{r}}}f(\mathbf{a})$ along $\hat{\mathbf{r}}$ exists at $\mathbf{a}$, then there exists a unique [vector](../../../../../Algebra/Linear%20Algebra/Matrices/Row%20and%20Column%20Vectors/Real%20Vectors/Real%20Vector.md) $\nabla f(\mathbf{a})$, which depends on $\mathbf{a}$ but not on $\hat{\mathbf{r}}$, such that $\partial_{\hat{\mathbf{r}}}f(\mathbf{a})$ is the [dot product](../../../../../Algebra/Linear%20Algebra/Matrices/Row%20and%20Column%20Vectors/Real%20Vectors/Real%20Dot%20Product.md) of $\nabla f(\mathbf{a})$ and $\hat{\mathbf{r}}$:
+>If $f$'s [directional derivative](Directional%20Derivatives%20of%20Real%20Scalar%20Fields.md) $\partial_{\hat{\mathbf{r}}}f(\mathbf{a})$ along every direction $\hat{\mathbf{r}}$ exists at $\mathbf{a}$, then there exists a unique [vector](../../../../../Algebra/Linear%20Algebra/Matrices/Row%20and%20Column%20Vectors/Real%20Vectors/Real%20Vector.md) $\nabla f(\mathbf{a})$, which depends on $\mathbf{a}$ but not on $\hat{\mathbf{r}}$, such that $\partial_{\hat{\mathbf{r}}}f(\mathbf{a})$ is the [dot product](../../../../../Algebra/Linear%20Algebra/Matrices/Row%20and%20Column%20Vectors/Real%20Vectors/Real%20Dot%20Product.md) of $\nabla f(\mathbf{a})$ and $\hat{\mathbf{r}}$:
 >
 >$$
 >(\nabla f(\mathbf{a})) \cdot \hat{\mathbf{r}} = \partial_{\hat{\mathbf{r}}}f(\mathbf{a})

vault/Mathematics/Analysis/Real Analysis/Multivariate Real Analysis/Vector Fields/Conservative Vector Fields/Conservative Vector Field.md

@@ -1,6 +1,6 @@
 >[!DEFINITION] Definition: Gradient Field
 >
->A [real vector field](../Real%20Vector%20Field.md) $\boldsymbol{v}: D \subseteq \mathbb{R}^n \to \mathbb{R}$ is **conservative** if there exists a [real scalar field](../../Scalar%20Fields/Real%20Scalar%20Field.md) $f: D \to \mathbb{R}$ whose [gradient](../../Scalar%20Fields/Differentiation/Gradient.md) is $\boldsymbol{v}$.
+>A [real vector field](../Real%20Vector%20Field.md) $\boldsymbol{v}: \mathcal{D} \subseteq \mathbb{R}^n \to \mathbb{R}^n$ is **conservative** iff there exists a [real scalar field](../../Scalar%20Fields/Real%20Scalar%20Field.md) $f: D \to \mathbb{R}$ whose [gradient](../../Scalar%20Fields/Differentiation/Gradient.md) is $\boldsymbol{v}$.
 >
 >$$
 >\nabla f = \boldsymbol{v}

vault/Mathematics/Analysis/Real Analysis/Multivariate Real Analysis/Vector Fields/Differentiation/Divergence.md

@@ -35,6 +35,22 @@
 >>
 >
 
+>[!THEOREM] Theorem: Divergence in Cartesian Coordinates
+>
+>Let $\boldsymbol{v}: \mathcal{D} \subseteq \mathbb{R}^n \to \mathbb{R}^n$ be a [real vector field](../Real%20Vector%20Field.md) with [component functions](../../Real%20Vector%20Functions/Real%20Vector%20Function.md) $v_1, \cdots, v_n$.
+>
+>If for every $i \in \{1, \dotsc, n\}$, the component function $v_i$ is [partially differentiable](../../Scalar%20Fields/Differentiation/Partial%20Differentiability%20of%20Real%20Scalar%20Fields.md) at $\mathbf{a} \in \mathcal{D}$ with respect to the $i$-th [Cartesian coordinate](../../../../../Geometry/Euclidean%20Geometry/Euclidean%20Space/Coordinate%20Systems/Cartesian%20Coordinate%20System.md) $x^i$, then the [divergence](Divergence.md) of $\boldsymbol{v}$ at $\mathbf{a}$ is given by
+>
+>$$
+>\nabla \cdot \boldsymbol{v}(\mathbf{a}) = \sum_{i = 1}^n \frac{\partial v_i}{\partial x^i}(\mathbf{a}) = \frac{\partial v_1}{\partial x^1}(\mathbf{a}) + \cdots + \frac{\partial v_n}{\partial x^n}(\mathbf{a})
+>$$
+>
+>>[!PROOF]-
+>>
+>>TODO
+>>
+>
+
 >[!THEOREM] Theorem: Linearity of the Divergence
 >
 >The [divergence](Divergence.md) is [linear transformation](../../../../../Algebra/Linear%20Algebra/Linear%20Transformations/Linear%20Transformation.md) - for all $\lambda, \gamma \in \mathbb{R}$ and all [vector fields](../Real%20Vector%20Field.md) $\boldsymbol{u}, \boldsymbol{v}: D \subseteq \mathbb{R}^n \to \mathbb{R}^n$ whose $i$-th [component functions](../../Real%20Vector%20Functions/Real%20Vector%20Function.md) are [partially differentiable](../../Scalar%20Fields/Differentiation/Partial%20Derivatives%20of%20Real%20Scalar%20Fields.md) with respect to the $i$-th variable:

vault/Mathematics/Geometry/Euclidean Geometry/Euclidean Space/Coordinate Systems/Cylindrical Coordinate System.md

@@ -1,6 +1,6 @@
 # Cylindrical Coordinate System
 
-Cylindrical coordinates are a natural extension of the [polar coordinate system](Polar%20Coordinate%20System.md) to three dimensions. They identify each point in the [Euclidean space](../Euclidean%20Space.md) $\mathbb{R}^3$ by its distance from the $z$-axis, the angle $\theta$ its projection in the $xy$-plane makes with the $x$-axis and its $z$ component.
+Cylindrical coordinates are a natural extension of the [polar coordinate system](Polar%20Coordinate%20System.md) to three dimensions. They identify each point in the [Euclidean space](../Euclidean%20Space.md) $\mathbb{R}^3$ by its distance from the $z$-axis, the angle $\phi$ its projection in the $xy$-plane makes with the $x$-axis and its $z$ component.
 
 ![](res/Cylindrical%20Coordinates.drawio.svg)
 
@@ -31,14 +31,16 @@ Cylindrical coordinates are a natural extension of the [polar coordinate system]
 
 ## Conventions
 
-A single point has infinitely many possible values for its azimuth, since adding or subtracting a multiple of $2\pi$ to an angle has no effect. To use the formalism of coordinate systems and insure uniqueness of the azimuthal angle for each point, however, one needs to restricts the possible values for it. Common conventions are for the range of $\phi$ are $[0; 2\pi)$ and $(-\pi, \pi]$.
+Some people prefer to use $\theta$ or $\varphi$ for the azimuth.
+
+Additionally, a single point has infinitely many possible values for its azimuth, since adding or subtracting a multiple of $2\pi$ to an angle has no effect. To use the formalism of coordinate systems and insure uniqueness of the azimuthal angle for each point, however, one needs to restricts the possible values for it. Common conventions are for the range of $\phi$ are $[0; 2\pi)$ and $(-\pi, \pi]$.
 
 >[!THEOREM] Theorem: Cylindrical Coordinate System
 >
 >The [function](../../../../Analysis/Real%20Analysis/Multivariate%20Real%20Analysis/Vector%20Fields/Real%20Vector%20Field.md) $s: \mathbb{R}^3 \to \mathbb{R}^3$ defined for each $\mathbf{p} = \begin{bmatrix}x & y & z\end{bmatrix}^\mathsf{T} \in \mathbb{R}^n$ as 
 >
 >$$
->s(\mathbf{p}) \overset{\text{def}}{=} (\rho, \theta, z) 
+>s(\mathbf{p}) \overset{\text{def}}{=} (\rho, \phi, z) 
 >$$
 >
 >where