# physics/pdv-to-frac-partial

Expand pdv forms to explicit partial-derivative fractions.

https://texform.dev/rules/physics/pdv-to-frac-partial

## Metadata

| Field | Value |
| --- | --- |
| Key | physics/pdv-to-frac-partial |
| Level | faithful |
| Fidelity | render |
| Enabled by packages | physics |
| Profiles | Faithful, Corpus, Equiv |

## Rewrite patterns

### pdv-two-argument

From:

```latex
\pdv{#1}{#2}
```

To:

```latex
\frac{\partial #1}{\partial #2}
```

### pdv-operator

From:

```latex
\pdv{#1}
```

To:

```latex
\frac{\partial}{\partial #1}
```

### pdv-order

From:

```latex
\pdv[#1]{#2}{#3}
```

To:

```latex
\frac{\partial^{#1} #2}{\partial #3^{#1}}
```

### pdv-mixed-partial

From:

```latex
\pdv{#1}{#2}{#3}
```

To:

```latex
\frac{\partial^{2} #1}{\partial #2 \partial #3}
```

### pdv-star

From:

```latex
\pdv*{#1}{#2}
```

To:

```latex
\flatfrac{\partial #1}{\partial #2}
```

## Examples

Examples show the rule in isolation, not a complete normalization profile.

### pdv embedded

Before:

```latex
\lambda=\pdv{f}{x}
```

After this rule:

```latex
\lambda=\frac{\partial f}{\partial x}
```

Packages: base, physics.

Rendering verified with the declared packages.

### pdv operator form

Before:

```latex
\pdv{x} f
```

After this rule:

```latex
\frac{\partial}{\partial x} f
```

Packages: base, physics.

Rendering verified with the declared packages.

### pdv second order

Before:

```latex
\pdv[2]{f}{x}
```

After this rule:

```latex
\frac{\partial^{2} f}{\partial x^{2}}
```

Packages: base, physics.

Rendering verified with the declared packages.

### pdv mixed partial

Before:

```latex
\pdv{f}{x}{y}
```

After this rule:

```latex
\frac{\partial^{2} f}{\partial x \partial y}
```

Packages: base, physics.

Rendering verified with the declared packages.

### pdv star flatfrac

Before:

```latex
\pdv*{f}{x}
```

After this rule:

```latex
\flatfrac{\partial f}{\partial x}
```

Packages: base, physics.

Rendering verified with the declared packages.

## Triggers

- [\\pdv](https://texform.dev/latex/commands/pdv)

## Eliminates

- [\\pdv](https://texform.dev/latex/commands/pdv)

## Produces

- [\\frac](https://texform.dev/latex/commands/frac)
- [\\flatfrac](https://texform.dev/latex/commands/flatfrac)
- [\\partial](https://texform.dev/latex/symbols/partial)

## Upstream

- [physics/partial-derivative-to-pdv](https://texform.dev/rules/physics/partial-derivative-to-pdv): Collapse long partial derivative names to the pdv command.

## Same family

- [physics/derivative-to-dv](https://texform.dev/rules/physics/derivative-to-dv): Collapse derivative to the compact dv command.
- [physics/functional-derivative-to-fdv](https://texform.dev/rules/physics/functional-derivative-to-fdv): Collapse long functional derivative names to the fdv command.
- [physics/partial-derivative-to-pdv](https://texform.dev/rules/physics/partial-derivative-to-pdv): Collapse long partial derivative names to the pdv command.
- [physics/vectorarrow-to-va](https://texform.dev/rules/physics/vectorarrow-to-va): Collapse vectorarrow to the short va helper.
- [physics/vectorbold-to-vb](https://texform.dev/rules/physics/vectorbold-to-vb): Collapse vectorbold to the short vb helper.
- [physics/vectorunit-to-vu](https://texform.dev/rules/physics/vectorunit-to-vu): Collapse vectorunit to the short vu helper.
- [physics/dv-to-frac-d](https://texform.dev/rules/physics/dv-to-frac-d): Expand dv forms to explicit d-based derivative fractions.
- [physics/fdv-to-frac-delta](https://texform.dev/rules/physics/fdv-to-frac-delta): Expand fdv forms to explicit delta-based derivative fractions.
- [physics/va-to-vec-mathbf](https://texform.dev/rules/physics/va-to-vec-mathbf): Canonicalize va to an explicit vec-wrapped mathbf form.
- [physics/vb-to-mathbf](https://texform.dev/rules/physics/vb-to-mathbf): Canonicalize vb to an explicit mathbf wrapper.
- [physics/vu-to-hat-mathbf](https://texform.dev/rules/physics/vu-to-hat-mathbf): Canonicalize vu to an explicit hat-wrapped mathbf form.

## Notes

Covers the two-argument partial derivative, one-argument operator form, optional [n] order, mixed partial third braced group, and star branch. For MathJax physics, a third braced group makes the numerator order depend on the number of denominator variables and ignores the optional [n] order; that combined form is intentionally not represented as a separate rewrite pattern here.

## Disable this rule

```python
TransformEngine(profile="corpus", disable_rules=["physics/pdv-to-frac-partial"])
```

```javascript
new TransformEngine({ profile: 'corpus', disableRules: ['physics/pdv-to-frac-partial'] })
```

[Open in Playground](https://play.texform.dev/?input=%5Clambda%3D%5Cpdv%7Bf%7D%7Bx%7D)