Tooltip Test Page
This page tests the tooltip functionality for math block references.
Basic Definitions
A vector space over a field $F$ is a set $V$ together with two operations: - Vector addition: $V \times V \to V$ - Scalar multiplication: $F \times V \to V$
satisfying the following axioms: 1. $(V, +)$ is an abelian group 2. Scalar multiplication is associative: $a(bv) = (ab)v$ 3. Distributive laws hold 4. Identity: $1v = v$ for all $v \in V$
Referenced by (3 direct, 32 transitive)
Direct references:
Transitive (depth 1):
- remark-9
- remark-16
- Hessian Matrix
- incompressible
- Bound vector
- Direction
- invariance-of-curl
- Jacobian Matrix
- Cross Product
- Unit Vector
- note-5
- Surface Normal Vector
- theorem-19
- remark-45
- Free vector
- remark-32
- Vector Multiplication by a Scalar
- note-3
Transitive (depth 2):
- Vector Equality
- proof-of-theorem-19
- remark-46
- Normal Derivative
- Directional Derivative
- Zero Vector
- remark-30
- directional-derivative-is-inner-product-of-vector-and-grad
- remark-23
- Tangent Vector
- gradient-as-surface-normal-vector
- proof-of-gradient-as-surface-normal-vector
Transitive (depth 3):
Let $V$ be a finite-dimensional vector space and $W$ be a subspace of $V$. Then: $$\dim(W) \leq \dim(V)$$ with equality if and only if $W = V$.
:::proof {label: dim-theorem-proof} Let $\{w_1, \ldots, w_k\}$ be a basis for $W$. Since $W \subseteq V$, these vectors are linearly independent in $V$. By the basis extension theorem, we can extend this to a basis of $V$.
Referenced by (1 direct)
Direct references:
Test References
Here are various types of references to test tooltips:
- Simple reference: A vector space over a field $F$...
- Typed reference: A vector space over a field $F$...
- Custom text reference: the definition of vector spaces
- Theorem reference: Dimension Theorem
- Another simple reference: Dimension Theorem
Cross-References from Other Pages
Let's also test some cross-file references if they exist:
- Reference to a definition that might exist elsewhere: @continuity
- Reference to a theorem that might exist elsewhere: @ftc
Nested Block Example
A subset $W$ of a vector space $V$ is a subspace if and only if: 1. $0 \in W$ 2. $W$ is closed under addition 3. $W$ is closed under scalar multiplication
:::example {label: symmetric-matrices-subspace} The set of all $n \times n$ symmetric matrices forms a subspace of $M_n(\mathbb{R})$.
Referenced by (1 direct)
Direct references:
Reference to the proposition: A subset $W$ of a vector space...
Invalid References
These should show error styling: - @nonexistent-label - @type:wrong-label
Math in Tooltips
The derivative of a function $f$ at a point $a$ is defined as: $$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$ provided this limit exists.
Referenced by (1 direct)
Direct references:
Reference with math: The derivative of a function $f$ at...