> ## Documentation Index
> Fetch the complete documentation index at: https://resources.latex-cloud-studio.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Physics Notation and Symbols

> Master physics notation in LaTeX - from quantum mechanics to relativity. Learn vectors, tensors, bra-ket notation, and specialized physics symbols.

export const RenderedOutput = ({title = "Rendered output", ctaHref, ctaLabel = "Open LaTeX Cloud Studio", children}) => {
  const [isExpanded, setIsExpanded] = useState(false);
  const trackEditorCta = () => {
    const target = new URL(ctaHref, window.location.href);
    globalThis.posthog?.capture?.("docs_app_cta_clicked", {
      source_page: window.location.pathname,
      source_section: "rendered_output",
      cta_variant: "first_compiled_example",
      target_url: target.toString(),
      target_utm_source: target.searchParams.get("utm_source"),
      target_utm_medium: target.searchParams.get("utm_medium"),
      target_utm_campaign: target.searchParams.get("utm_campaign"),
      target_utm_content: target.searchParams.get("utm_content")
    }, {
      transport: "sendBeacon",
      send_instantly: true
    });
  };
  return <details className="rendered-output" onToggle={event => setIsExpanded(event.currentTarget.open)}>
      <summary className="rendered-output__summary">
        <span className="rendered-output__title">{title}</span>
        <span className="rendered-output__hint" aria-hidden="true">View compiled result</span>
      </summary>
      {isExpanded && <div className="rendered-output__content">
          {children}
          {ctaHref && <aside className="rendered-output__cta" aria-label="Continue in the LaTeX editor">
              <span>
                <strong>Ready to use this syntax?</strong>
                Continue in the browser editor when you want to adapt the example in a real project.
              </span>
              <a href={ctaHref} onClick={trackEditorCta}>{ctaLabel}<span aria-hidden="true"> →</span></a>
            </aside>}
        </div>}
    </details>;
};

export const LatexSource = ({filename, source}) => {
  const [copyStatus, setCopyStatus] = useState("Copy");
  const copySource = async () => {
    try {
      await navigator.clipboard.writeText(source);
      setCopyStatus("Copied");
    } catch {
      setCopyStatus("Select and copy");
    }
  };
  return <figure className="latex-source">
      <figcaption className="latex-source__header">
        <span className="latex-source__filename">{filename}</span>
        <button type="button" className="latex-source__copy" onClick={copySource} aria-live="polite">
          {copyStatus}
        </button>
      </figcaption>
      <pre className="latex-source__pre" aria-label={`LaTeX source: ${filename}`} tabIndex="0">
        <code className="language-latex">{source}</code>
      </pre>
    </figure>;
};

export const LatexPreview = ({src, alt, caption, width, height}) => {
  const minZoom = 1;
  const maxZoom = 3;
  const zoomStep = 0.5;
  const measureSvgContent = async (assetSrc, pageWidth, pageHeight) => {
    const cacheKey = "__latexCloudSvgContentBoxCache";
    const contentBoxCache = globalThis[cacheKey] ?? new Map();
    globalThis[cacheKey] = contentBoxCache;
    if (contentBoxCache.has(assetSrc)) return contentBoxCache.get(assetSrc);
    const measurement = (async () => {
      const assetUrl = new URL(assetSrc, window.location.href);
      if (assetUrl.origin !== window.location.origin) {
        throw new Error("Rendered output must use a same-origin SVG asset.");
      }
      const response = await fetch(assetUrl, {
        credentials: "same-origin"
      });
      if (!response.ok) throw new Error(`Rendered output request failed with ${response.status}.`);
      const source = await response.text();
      const documentNode = new DOMParser().parseFromString(source, "image/svg+xml");
      if (documentNode.querySelector("parsererror")) throw new Error("Rendered output is not valid SVG.");
      const sourceSvg = documentNode.documentElement;
      sourceSvg.querySelectorAll("script, foreignObject").forEach(node => node.remove());
      [sourceSvg, ...sourceSvg.querySelectorAll("*")].forEach(node => {
        [...node.attributes].forEach(attribute => {
          if ((/^on/i).test(attribute.name)) node.removeAttribute(attribute.name);
          if ((attribute.name === "href" || attribute.name === "xlink:href") && !attribute.value.startsWith("#")) {
            node.removeAttribute(attribute.name);
          }
        });
      });
      const measurementHost = document.createElement("div");
      measurementHost.className = "latex-preview__measurement-host";
      const measuredSvg = document.importNode(sourceSvg, true);
      measuredSvg.setAttribute("aria-hidden", "true");
      measurementHost.appendChild(measuredSvg);
      document.body.appendChild(measurementHost);
      try {
        const measuredElements = [...measuredSvg.children].filter(node => !["defs", "desc", "metadata", "style", "title"].includes(node.tagName.toLowerCase()));
        const elementBounds = measuredElements.map(node => node.getBBox()).filter(box => [box.x, box.y, box.width, box.height].every(Number.isFinite) && box.width > 0 && box.height > 0);
        if (elementBounds.length === 0) {
          throw new Error("Rendered output has no measurable visible content.");
        }
        const sortedBounds = [...elementBounds].sort((left, right) => left.y - right.y);
        const clusterGap = pageHeight * 0.045;
        const clusters = [];
        sortedBounds.forEach(box => {
          const current = clusters[clusters.length - 1];
          if (!current || box.y - current.bottom > clusterGap) {
            clusters.push({
              boxes: [box],
              bottom: box.y + box.height
            });
            return;
          }
          current.boxes.push(box);
          current.bottom = Math.max(current.bottom, box.y + box.height);
        });
        const contentClusters = clusters.filter(cluster => {
          const clusterBox = cluster.boxes.reduce((combined, box) => {
            const right = Math.max(combined.x + combined.width, box.x + box.width);
            const bottom = Math.max(combined.y + combined.height, box.y + box.height);
            const x = Math.min(combined.x, box.x);
            const y = Math.min(combined.y, box.y);
            return {
              x,
              y,
              width: right - x,
              height: bottom - y
            };
          });
          const centerY = clusterBox.y + clusterBox.height / 2;
          const isMarginFurniture = cluster.boxes.length <= 2 && clusterBox.width < pageWidth * 0.2 && clusterBox.height < pageHeight * 0.04 && (centerY < pageHeight * 0.08 || centerY > pageHeight * 0.8);
          return !isMarginFurniture;
        });
        const visibleBounds = (contentClusters.length > 0 ? contentClusters : clusters).flatMap(cluster => cluster.boxes);
        const bounds = visibleBounds.reduce((combined, box) => {
          const right = Math.max(combined.x + combined.width, box.x + box.width);
          const bottom = Math.max(combined.y + combined.height, box.y + box.height);
          const x = Math.min(combined.x, box.x);
          const y = Math.min(combined.y, box.y);
          return {
            x,
            y,
            width: right - x,
            height: bottom - y
          };
        });
        const clampValue = (value, minimum, maximum) => Math.min(maximum, Math.max(minimum, value));
        const padding = Math.max(8, Math.min(pageWidth, pageHeight) * 0.025);
        const x = clampValue(bounds.x - padding, 0, pageWidth);
        const y = clampValue(bounds.y - padding, 0, pageHeight);
        const right = clampValue(bounds.x + bounds.width + padding, 0, pageWidth);
        const bottom = clampValue(bounds.y + bounds.height + padding, 0, pageHeight);
        return {
          x,
          y,
          width: right - x,
          height: bottom - y
        };
      } finally {
        measurementHost.remove();
      }
    })();
    contentBoxCache.set(assetSrc, measurement);
    measurement.catch(() => contentBoxCache.delete(assetSrc));
    return measurement;
  };
  const renderPreviewAsset = ({contentBox: assetContentBox, loading}) => {
    if (!assetContentBox) {
      return <img className="latex-preview__asset" src={src} alt={alt} width={width} height={height} loading={loading} draggable="false" />;
    }
    return <svg className="latex-preview__asset" viewBox={`${assetContentBox.x} ${assetContentBox.y} ${assetContentBox.width} ${assetContentBox.height}`} preserveAspectRatio="xMidYMid meet" role="img" aria-label={alt}>
        <image href={src} x="0" y="0" width={width} height={height} />
      </svg>;
  };
  const [isOpen, setIsOpen] = useState(false);
  const [frameMode, setFrameMode] = useState("content");
  const [viewMode, setViewMode] = useState("fit");
  const [zoom, setZoom] = useState(minZoom);
  const [contentBox, setContentBox] = useState(null);
  const [measurementStatus, setMeasurementStatus] = useState("loading");
  const dialogRef = useRef(null);
  const closeButtonRef = useRef(null);
  const viewportRef = useRef(null);
  const previousFocusRef = useRef(null);
  const dragRef = useRef(null);
  useEffect(() => {
    let isCurrent = true;
    setMeasurementStatus("loading");
    measureSvgContent(src, width, height).then(box => {
      if (!isCurrent) return;
      setContentBox(box);
      setMeasurementStatus("ready");
    }).catch(() => {
      if (!isCurrent) return;
      setContentBox(null);
      setFrameMode("page");
      setMeasurementStatus("error");
    });
    return () => {
      isCurrent = false;
    };
  }, [height, src, width]);
  const closeViewer = useCallback(() => {
    setIsOpen(false);
  }, []);
  const openViewer = () => {
    previousFocusRef.current = document.activeElement;
    setFrameMode(contentBox ? "content" : "page");
    setViewMode("fit");
    setZoom(minZoom);
    setIsOpen(true);
  };
  const applyZoom = useCallback(nextZoom => {
    const boundedZoom = Math.min(maxZoom, Math.max(minZoom, nextZoom));
    setViewMode("custom");
    setZoom(boundedZoom);
  }, []);
  const zoomIn = useCallback(() => {
    applyZoom(viewMode === "fit" ? minZoom + zoomStep : zoom + zoomStep);
  }, [applyZoom, viewMode, zoom]);
  const zoomOut = useCallback(() => {
    applyZoom(viewMode === "fit" ? minZoom : zoom - zoomStep);
  }, [applyZoom, viewMode, zoom]);
  useEffect(() => {
    if (!isOpen) return undefined;
    const previousOverflow = document.body.style.overflow;
    document.body.style.overflow = "hidden";
    closeButtonRef.current?.focus();
    return () => {
      document.body.style.overflow = previousOverflow;
      previousFocusRef.current?.focus?.();
    };
  }, [isOpen]);
  useEffect(() => {
    if (!isOpen) return undefined;
    const handleKeyDown = event => {
      if (event.key === "Escape") {
        event.preventDefault();
        closeViewer();
        return;
      }
      if ((event.key === "+" || event.key === "=") && !event.metaKey && !event.ctrlKey) {
        event.preventDefault();
        zoomIn();
        return;
      }
      if (event.key === "-" && !event.metaKey && !event.ctrlKey) {
        event.preventDefault();
        zoomOut();
        return;
      }
      if (event.key !== "Tab" || !dialogRef.current) return;
      const focusable = [...dialogRef.current.querySelectorAll('button:not([disabled]), a[href], [tabindex]:not([tabindex="-1"])')];
      if (focusable.length === 0) return;
      const first = focusable[0];
      const last = focusable[focusable.length - 1];
      if (event.shiftKey && document.activeElement === first) {
        event.preventDefault();
        last.focus();
      } else if (!event.shiftKey && document.activeElement === last) {
        event.preventDefault();
        first.focus();
      }
    };
    window.addEventListener("keydown", handleKeyDown);
    return () => {
      window.removeEventListener("keydown", handleKeyDown);
    };
  }, [closeViewer, isOpen, zoomIn, zoomOut]);
  const startDrag = event => {
    if (event.button !== 0 || !viewportRef.current) return;
    const viewport = viewportRef.current;
    dragRef.current = {
      pointerId: event.pointerId,
      x: event.clientX,
      y: event.clientY,
      scrollLeft: viewport.scrollLeft,
      scrollTop: viewport.scrollTop
    };
    viewport.setPointerCapture(event.pointerId);
    viewport.dataset.dragging = "true";
  };
  const continueDrag = event => {
    const drag = dragRef.current;
    const viewport = viewportRef.current;
    if (!drag || !viewport || drag.pointerId !== event.pointerId) return;
    viewport.scrollLeft = drag.scrollLeft - (event.clientX - drag.x);
    viewport.scrollTop = drag.scrollTop - (event.clientY - drag.y);
  };
  const stopDrag = event => {
    const viewport = viewportRef.current;
    if (viewport?.hasPointerCapture(event.pointerId)) viewport.releasePointerCapture(event.pointerId);
    if (viewport) delete viewport.dataset.dragging;
    dragRef.current = null;
  };
  const activeContentBox = frameMode === "content" ? contentBox : null;
  const activeWidth = activeContentBox?.width ?? width;
  const activeHeight = activeContentBox?.height ?? height;
  const activeRatio = activeWidth / activeHeight;
  const inlineContentBox = measurementStatus === "ready" ? contentBox : null;
  const inlineWidth = inlineContentBox?.width ?? width;
  const inlineHeight = inlineContentBox?.height ?? height;
  const inlineGeometry = {
    aspectRatio: `${inlineWidth} / ${inlineHeight}`,
    maxWidth: `${30 * inlineWidth / inlineHeight}rem`
  };
  const imageStyle = viewMode === "fit" ? {
    aspectRatio: `${activeWidth} / ${activeHeight}`,
    width: "100%",
    maxWidth: `${Math.max(16, activeRatio * 78)}dvh`
  } : {
    aspectRatio: `${activeWidth} / ${activeHeight}`,
    width: `${zoom * 100}%`,
    maxWidth: "none"
  };
  const zoomLabel = viewMode === "fit" ? frameMode === "content" ? "Fit content" : "Full page" : `${Math.round(zoom * 100)}%`;
  return <figure className="latex-preview">
      <button type="button" className="latex-preview__trigger" onClick={openViewer} aria-haspopup="dialog" aria-label={`Open zoomable preview: ${alt}`}>
        <span className="latex-preview__page" style={inlineGeometry}>
          {measurementStatus === "loading" ? <span className="latex-preview__loading" role="status">Preparing compiled output…</span> : renderPreviewAsset({
    src,
    alt,
    width,
    height,
    contentBox: inlineContentBox,
    loading: "lazy"
  })}
        </span>
        <span className="latex-preview__trigger-label" aria-hidden="true">
          <span className="latex-preview__trigger-icon">⌕</span>
          Open viewer
        </span>
      </button>
      <figcaption className="latex-preview__caption">
        <span>
          {caption}
          {measurementStatus === "error" && <span className="latex-preview__status" role="status"> Content fit is unavailable; the complete vector page is shown.</span>}
        </span>
        <a href={src} target="_blank" rel="noreferrer" className="latex-preview__source-link">Open SVG</a>
      </figcaption>

      {isOpen && <div className="latex-preview__backdrop" onMouseDown={event => {
    if (event.target === event.currentTarget) closeViewer();
  }}>
          <section ref={dialogRef} className="latex-preview__dialog" role="dialog" aria-modal="true" aria-label={`Rendered LaTeX viewer: ${alt}`}>
            <header className="latex-preview__toolbar">
              <div className="latex-preview__identity">
                <span className="latex-preview__eyebrow">Compiled LaTeX</span>
                <span className="latex-preview__filename">{alt}</span>
              </div>
              <div className="latex-preview__controls" aria-label="Preview controls">
                <button type="button" className={frameMode === "content" && viewMode === "fit" ? "is-active" : undefined} disabled={!contentBox} onClick={() => {
    setFrameMode("content");
    setViewMode("fit");
    setZoom(minZoom);
  }}>
                  Fit content
                </button>
                <button type="button" className={frameMode === "page" && viewMode === "fit" ? "is-active" : undefined} onClick={() => {
    setFrameMode("page");
    setViewMode("fit");
    setZoom(minZoom);
  }}>
                  Full page
                </button>
                <span className="latex-preview__zoom-group">
                  <button type="button" onClick={zoomOut} disabled={viewMode === "fit" || zoom <= minZoom} aria-label="Zoom out">−</button>
                  <output aria-live="polite" aria-label="Current zoom">{zoomLabel}</output>
                  <button type="button" onClick={zoomIn} disabled={viewMode !== "fit" && zoom >= maxZoom} aria-label="Zoom in">+</button>
                </span>
                <a href={src} target="_blank" rel="noreferrer">Open SVG</a>
                <button ref={closeButtonRef} type="button" className="latex-preview__close" onClick={closeViewer} aria-label="Close rendered LaTeX viewer">
                  Close
                </button>
              </div>
            </header>
            <div ref={viewportRef} className="latex-preview__viewport" data-view-mode={viewMode} onPointerDown={startDrag} onPointerMove={continueDrag} onPointerUp={stopDrag} onPointerCancel={stopDrag}>
              <span className="latex-preview__dialog-page" style={imageStyle}>
                {renderPreviewAsset({
    src,
    alt,
    width,
    height,
    contentBox: activeContentBox
  })}
              </span>
            </div>
            <footer className="latex-preview__viewer-note">
              Compiler-generated vector output · Use +/− to zoom · Drag to pan · Esc to close
            </footer>
          </section>
        </div>}
    </figure>;
};

Learn how to typeset physics equations, symbols, and notation professionally in LaTeX.

## Essential Physics Packages

<LatexSource filename="example.tex" source={"\\usepackage{amsmath}     % Essential math\n\\usepackage{amssymb}     % Extra symbols\n\\usepackage{physics}     % Physics shortcuts\n\\usepackage{siunitx}     % SI units\n\\usepackage{tensor}      % Tensor notation\n\\usepackage{braket}      % Quantum mechanics"} />

<RenderedOutput title="Expected effect">
  <Info>
    This is setup or structural LaTeX code. It changes available commands or document behavior, but it does not produce meaningful standalone page content by itself.
  </Info>
</RenderedOutput>

## Common Physics Symbols

| Symbol | LaTeX      | Description             |
| :----: | :--------- | :---------------------- |
|  **ℏ** | `\hbar`    | Reduced Planck constant |
|  **∇** | `\nabla`   | Gradient/Del operator   |
|  **∂** | `\partial` | Partial derivative      |
|  **∞** | `\infty`   | Infinity                |
|  **·** | `\cdot`    | Dot product             |
|  **×** | `\times`   | Cross product           |
|  **⊗** | `\otimes`  | Tensor product          |
|  **∑** | `\sum`     | Summation               |
|  **∫** | `\int`     | Integral                |
|  **∮** | `\oint`    | Closed integral         |

## Vectors and Tensors

### Vector Notation

<LatexSource filename="example.tex" source={"% Different vector styles\n\\vec{F} = m\\vec{a}              % Arrow notation\n\\mathbf{F} = m\\mathbf{a}        % Bold notation\n\\boldsymbol{\\tau} = \\vec{r} \\times \\vec{F}  % Bold Greek\n\n% Vector operations\n\\vec{A} \\cdot \\vec{B}           % Dot product\n\\vec{A} \\times \\vec{B}          % Cross product\n|\\vec{v}| \\text{ or } \\|\\vec{v}\\|  % Magnitude\n\n% Unit vectors\n\\hat{i}, \\hat{j}, \\hat{k}       % Cartesian\n\\hat{r}, \\hat{\\theta}, \\hat{\\phi}  % Spherical"} />

<RenderedOutput title="Expected effect">
  <Info>
    This code is a contextual or intentionally partial LaTeX excerpt, not a self-contained compilable document. Its visible result depends on the surrounding document, so the documentation shows the expected role without inventing a standalone preview.
  </Info>
</RenderedOutput>

<Card title="Expected output" icon="eye">
  **Newton's Second Law:** $\vec{F} = m\vec{a}$

  **Dot Product:** $\vec{A} \cdot \vec{B}$

  **Cross Product:** $\vec{A} \times \vec{B}$

  **Unit Vectors:** $\hat{i}, \hat{j}, \hat{k}$
</Card>

### Tensor Notation

<LatexSource filename="example.tex" source={"% Tensor indices\nT^{\\mu\\nu}                      % Contravariant\nT_{\\mu\\nu}                      % Covariant\nT^{\\mu}_{\\nu}                   % Mixed\n\n% Einstein notation\ng_{\\mu\\nu} = \\eta_{\\mu\\nu} + h_{\\mu\\nu}\n\n% Christoffel symbols\n\\Gamma^{\\lambda}_{\\mu\\nu}\n\n% Riemann tensor\nR^{\\rho}_{\\sigma\\mu\\nu}"} />

<RenderedOutput title="Expected effect">
  <Info>
    This code is a contextual or intentionally partial LaTeX excerpt, not a self-contained compilable document. Its visible result depends on the surrounding document, so the documentation shows the expected role without inventing a standalone preview.
  </Info>
</RenderedOutput>

## Quantum Mechanics

### Bra-Ket Notation

<LatexSource filename="example.tex" source={"% Basic bra-ket\n\\ket{\\psi}                      % Ket\n\\bra{\\phi}                      % Bra\n\\braket{\\phi|\\psi}              % Inner product\n\\braket{\\phi|H|\\psi}            % Matrix element\n\n% Operators\n\\hat{H}\\ket{\\psi} = E\\ket{\\psi} % Eigenvalue equation\n\\hat{p} = -i\\hbar\\frac{\\partial}{\\partial x}\n\n% Commutators\n[\\hat{x}, \\hat{p}] = i\\hbar\n\\{\\hat{a}, \\hat{a}^{\\dagger}\\} = 1  % Anticommutator"} />

<RenderedOutput title="Expected effect">
  <Info>
    This code is a contextual or intentionally partial LaTeX excerpt, not a self-contained compilable document. Its visible result depends on the surrounding document, so the documentation shows the expected role without inventing a standalone preview.
  </Info>
</RenderedOutput>

### Wave Functions

<LatexSource filename="example.tex" source={"% Schrödinger equation\ni\\hbar\\frac{\\partial}{\\partial t}\\Psi = \\hat{H}\\Psi\n\n% Plane wave\n\\psi(x,t) = Ae^{i(kx - \\omega t)}\n\n% Spherical harmonics\nY_{\\ell}^m(\\theta, \\phi)\n\n% Probability density\n|\\psi(x,t)|^2 = \\psi^*(x,t)\\psi(x,t)"} />

<RenderedOutput title="Expected effect">
  <Info>
    This code is a contextual or intentionally partial LaTeX excerpt, not a self-contained compilable document. Its visible result depends on the surrounding document, so the documentation shows the expected role without inventing a standalone preview.
  </Info>
</RenderedOutput>

## Classical Mechanics

### Lagrangian and Hamiltonian

<LatexSource filename="example.tex" source={"% Lagrangian\nL = T - V = \\frac{1}{2}m\\dot{x}^2 - V(x)\n\n% Euler-Lagrange equation\n\\frac{d}{dt}\\frac{\\partial L}{\\partial \\dot{q}} - \\frac{\\partial L}{\\partial q} = 0\n\n% Hamiltonian\nH = \\sum_i p_i\\dot{q}_i - L\n\n% Hamilton's equations\n\\dot{q}_i = \\frac{\\partial H}{\\partial p_i}, \\quad\n\\dot{p}_i = -\\frac{\\partial H}{\\partial q_i}"} />

<RenderedOutput title="Expected effect">
  <Info>
    This code is a contextual or intentionally partial LaTeX excerpt, not a self-contained compilable document. Its visible result depends on the surrounding document, so the documentation shows the expected role without inventing a standalone preview.
  </Info>
</RenderedOutput>

## Electromagnetism

### Maxwell's Equations

<LatexSource filename="example.tex" source={"% Differential form\n\\nabla \\cdot \\vec{E} = \\frac{\\rho}{\\epsilon_0}\n\\nabla \\cdot \\vec{B} = 0\n\\nabla \\times \\vec{E} = -\\frac{\\partial \\vec{B}}{\\partial t}\n\\nabla \\times \\vec{B} = \\mu_0\\vec{J} + \\mu_0\\epsilon_0\\frac{\\partial \\vec{E}}{\\partial t}\n\n% Integral form\n\\oint_S \\vec{E} \\cdot d\\vec{A} = \\frac{Q_{\\text{enc}}}{\\epsilon_0}\n\\oint_C \\vec{E} \\cdot d\\vec{\\ell} = -\\frac{d\\Phi_B}{dt}"} />

<RenderedOutput title="Expected effect">
  <Info>
    This code is a contextual or intentionally partial LaTeX excerpt, not a self-contained compilable document. Its visible result depends on the surrounding document, so the documentation shows the expected role without inventing a standalone preview.
  </Info>
</RenderedOutput>

### Field Notation

<LatexSource filename="example.tex" source={"% Electromagnetic tensor\nF^{\\mu\\nu} = \\partial^{\\mu}A^{\\nu} - \\partial^{\\nu}A^{\\mu}\n\n% Four-potential\nA^{\\mu} = (\\phi/c, \\vec{A})\n\n% Lorentz force\n\\vec{F} = q(\\vec{E} + \\vec{v} \\times \\vec{B})"} />

<RenderedOutput title="Expected effect">
  <Info>
    This code is a contextual or intentionally partial LaTeX excerpt, not a self-contained compilable document. Its visible result depends on the surrounding document, so the documentation shows the expected role without inventing a standalone preview.
  </Info>
</RenderedOutput>

## Thermodynamics

### Thermodynamic Relations

<LatexSource filename="example.tex" source={"% First law\ndU = \\delta Q - \\delta W = TdS - PdV\n\n% Partial derivatives\n\\left(\\frac{\\partial U}{\\partial S}\\right)_V = T\n\\left(\\frac{\\partial U}{\\partial V}\\right)_S = -P\n\n% Maxwell relations\n\\left(\\frac{\\partial T}{\\partial V}\\right)_S = -\\left(\\frac{\\partial P}{\\partial S}\\right)_V"} />

<RenderedOutput title="Expected effect">
  <Info>
    This code is a contextual or intentionally partial LaTeX excerpt, not a self-contained compilable document. Its visible result depends on the surrounding document, so the documentation shows the expected role without inventing a standalone preview.
  </Info>
</RenderedOutput>

## Units and Constants

### SI Units with siunitx

<LatexSource filename="example.tex" source={"% Basic units\n\\SI{3e8}{m/s}                   % Speed of light\n\\SI{6.626e-34}{J.s}             % Planck constant\n\\SI{9.81}{m/s^2}                % Acceleration\n\n% Complex units\n\\SI{13.6}{eV}                   % Energy\n\\SI{2.5}{kg.m/s}                % Momentum\n\\SI{1.23e-4}{N.m}               % Torque\n\n% Uncertainties\n\\SI{9.81 \\pm 0.02}{m/s^2}"} />

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## Special Relativity

### Four-Vectors and Metrics

<LatexSource filename="example.tex" source={"% Four-vectors\nx^{\\mu} = (ct, \\vec{x})\np^{\\mu} = (E/c, \\vec{p})\n\n% Minkowski metric\n\\eta_{\\mu\\nu} = \\text{diag}(1, -1, -1, -1)\n\n% Lorentz transformation\nx'^{\\mu} = \\Lambda^{\\mu}_{\\nu} x^{\\nu}\n\n% Invariant interval\nds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2"} />

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## Tips for Physics Documents

<CardGroup cols={2}>
  <Card title="Use Consistent Notation" icon="check">
    Stick to either arrows or bold for vectors throughout your document
  </Card>

  <Card title="Define Your Symbols" icon="book">
    Always define non-standard symbols when first introduced
  </Card>

  <Card title="Group Related Equations" icon="object-group">
    Use align environments for related equations
  </Card>

  <Card title="Number Important Equations" icon="hashtag">
    Only number equations you reference later
  </Card>
</CardGroup>

## Common Physics Environments

<LatexSource filename="example.tex" source={"% For derivations\n\\begin{align}\n  F &= ma \\\\\n  &= m\\frac{dv}{dt} \\\\\n  &= \\frac{dp}{dt}\n\\end{align}\n\n% For definitions\n\\begin{equation}\n  \\boxed{E = mc^2}\n\\end{equation}\n\n% For multiple cases\n\\begin{cases}\n  \\psi(x) = Ae^{ikx} + Be^{-ikx} & \\text{for } x < 0 \\\\\n  \\psi(x) = Ce^{-\\kappa x} & \\text{for } x > 0\n\\end{cases}"} />

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<Info>
  **Pro tip**: The `physics` package provides many shortcuts like `\dv{f}{x}` for derivatives and `\abs{\psi}` for absolute values.
</Info>

## Further Reading

<CardGroup cols={2}>
  <Card title="Mathematical Expressions" icon="square-root-variable" href="/learn/latex/mathematics/mathematical-expressions">
    General mathematical typesetting
  </Card>

  <Card title="Chemistry Notation" icon="flask" href="/learn/latex/specialized-notation/chemistry">
    Chemical formulas and equations
  </Card>

  <Card title="Symbol Reference" icon="book" href="/learn/reference/symbols">
    Complete symbol reference
  </Card>

  <Card title="Advanced Math" icon="function" href="/learn/latex/mathematics/advanced-math">
    Advanced mathematical concepts
  </Card>
</CardGroup>