> ## 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.

# Advanced Mathematics

> Explore advanced mathematical typesetting in LaTeX. Master complex analysis, abstract algebra, topology, and professional theorem environments.

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>;
};

Take your mathematical typesetting to the next level with advanced LaTeX techniques for complex mathematics.

## Advanced Mathematical Packages

<LatexSource filename="example.tex" source={"\\usepackage{amsmath}        % Essential math enhancements\n\\usepackage{amssymb}        % Additional symbols\n\\usepackage{amsthm}         % Theorem environments\n\\usepackage{mathtools}      % Extensions to amsmath\n\\usepackage{tensor}         % Tensor notation\n\\usepackage{mathrsfs}       % Script letters\n\\usepackage{bbm}            % Blackboard bold\n\\usepackage{dsfont}         % Double stroke fonts"} />

<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>

## Abstract Algebra

### Groups and Rings

<LatexSource filename="example.tex" source={"% Group theory\nG = \\langle a, b \\mid a^2 = b^3 = (ab)^2 = e \\rangle\n\n% Quotient groups\nG/H \\cong \\mathbb{Z}_n\n\n% Ring homomorphism\n\\phi: R \\to S, \\quad \\phi(xy) = \\phi(x)\\phi(y)\n\n% Ideals\n\\mathfrak{p} \\triangleleft R\n\n% Galois groups\n\\text{Gal}(K/F) = \\text{Aut}(K/F)"} />

<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 Extensions

<LatexSource filename="example.tex" source={"% Field extension degree\n[K : F] = \\dim_F K\n\n% Algebraic closure\n\\overline{\\mathbb{Q}}\n\n% Splitting field\nK = F(\\alpha_1, \\ldots, \\alpha_n)\n\n% Minimal polynomial\nm_{\\alpha,F}(x) = \\text{irr}(\\alpha, F)"} />

<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>

## Complex Analysis

### Complex Functions

<LatexSource filename="example.tex" source={"% Holomorphic function\nf: \\mathbb{C} \\to \\mathbb{C} \\text{ holomorphic}\n\n% Cauchy-Riemann equations\n\\frac{\\partial u}{\\partial x} = \\frac{\\partial v}{\\partial y}, \\quad\n\\frac{\\partial u}{\\partial y} = -\\frac{\\partial v}{\\partial x}\n\n% Residue theorem\n\\oint_{\\gamma} f(z) \\, dz = 2\\pi i \\sum_{k} \\text{Res}(f, z_k)\n\n% Laurent series\nf(z) = \\sum_{n=-\\infty}^{\\infty} a_n (z - z_0)^n"} />

<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>

### Contour Integration

<LatexSource filename="example.tex" source={"% Complex integral\n\\int_{\\gamma} f(z) \\, dz = \\int_a^b f(\\gamma(t)) \\gamma'(t) \\, dt\n\n% Winding number\nn(\\gamma, z_0) = \\frac{1}{2\\pi i} \\oint_{\\gamma} \\frac{dz}{z - z_0}\n\n% Branch cuts\n\\log z = \\ln|z| + i\\arg(z), \\quad -\\pi < \\arg(z) \\leq \\pi"} />

<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>

## Topology

### Topological Spaces

<LatexSource filename="example.tex" source={"% Open sets\n\\tau = \\{U \\subseteq X : U \\text{ is open}\\}\n\n% Closure and interior\n\\overline{A} = \\bigcap\\{F : A \\subseteq F, F \\text{ closed}\\}\nA^{\\circ} = \\bigcup\\{U : U \\subseteq A, U \\text{ open}\\}\n\n% Continuity\nf^{-1}(V) \\in \\tau_X \\text{ for all } V \\in \\tau_Y\n\n% Compactness\nX = \\bigcup_{i \\in I} U_i \\implies X = \\bigcup_{j=1}^n U_{i_j}"} />

<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>

### Algebraic Topology

<LatexSource filename="example.tex" source={"% Fundamental group\n\\pi_1(X, x_0) = \\{[\\gamma] : \\gamma \\text{ loop at } x_0\\}\n\n% Homology groups\nH_n(X) = \\ker(\\partial_n) / \\text{im}(\\partial_{n+1})\n\n% Euler characteristic\n\\chi(X) = \\sum_{i=0}^{\\infty} (-1)^i \\text{rank}(H_i(X))\n\n% Covering spaces\np: \\tilde{X} \\to X \\text{ covering map}"} />

<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>

## Category Theory

### Categories and Functors

<LatexSource filename="example.tex" source={"% Category\n\\mathcal{C} = (\\text{Ob}(\\mathcal{C}), \\text{Mor}(\\mathcal{C}))\n\n% Morphisms\nf: A \\to B \\in \\text{Hom}_{\\mathcal{C}}(A, B)\n\n% Functors\nF: \\mathcal{C} \\to \\mathcal{D}\n\n% Natural transformation\n\\eta: F \\Rightarrow G\n\n% Commutative diagram\n\\begin{tikzcd}\nA \\arrow[r, \"f\"] \\arrow[d, \"g\"'] & B \\arrow[d, \"h\"] \\\\\nC \\arrow[r, \"k\"'] & D\n\\end{tikzcd}"} />

<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>

## Number Theory

### Algebraic Number Theory

<LatexSource filename="example.tex" source={"% Ring of integers\n\\mathcal{O}_K = \\{x \\in K : x \\text{ integral over } \\mathbb{Z}\\}\n\n% Norm and trace\nN_{K/\\mathbb{Q}}(\\alpha) = \\prod_{i=1}^n \\sigma_i(\\alpha)\n\\text{Tr}_{K/\\mathbb{Q}}(\\alpha) = \\sum_{i=1}^n \\sigma_i(\\alpha)\n\n% Class number\nh_K = |\\text{Cl}(K)|\n\n% Dedekind zeta function\n\\zeta_K(s) = \\sum_{\\mathfrak{a}} \\frac{1}{N(\\mathfrak{a})^s}"} />

<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>

### Analytic Number Theory

<LatexSource filename="example.tex" source={"% Riemann zeta function\n\\zeta(s) = \\sum_{n=1}^{\\infty} \\frac{1}{n^s} = \\prod_p \\frac{1}{1 - p^{-s}}\n\n% Dirichlet L-functions\nL(s, \\chi) = \\sum_{n=1}^{\\infty} \\frac{\\chi(n)}{n^s}\n\n% Prime number theorem\n\\pi(x) \\sim \\frac{x}{\\ln x}\n\n% Möbius function\n\\mu(n) = \\begin{cases}\n1 & n = 1 \\\\\n(-1)^k & n = p_1 \\cdots p_k \\\\\n0 & \\text{otherwise}\n\\end{cases}"} />

<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>

## Differential Geometry

### Manifolds

<LatexSource filename="example.tex" source={"% Tangent space\nT_p M = \\{v : C^{\\infty}(M) \\to \\mathbb{R} \\mid v \\text{ derivation at } p\\}\n\n% Differential forms\n\\omega \\in \\Omega^k(M)\n\n% Exterior derivative\nd\\omega = \\sum_{i_0 < \\cdots < i_k} \\sum_{j} \\frac{\\partial f_{i_0\\ldots i_k}}{\\partial x^j} dx^j \\wedge dx^{i_0} \\wedge \\cdots \\wedge dx^{i_k}\n\n% Lie derivative\n\\mathcal{L}_X \\omega = d(i_X \\omega) + i_X(d\\omega)"} />

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### Riemannian Geometry

<LatexSource filename="example.tex" source={"% Metric tensor\ng_{ij} = \\langle \\frac{\\partial}{\\partial x^i}, \\frac{\\partial}{\\partial x^j} \\rangle\n\n% Christoffel symbols\n\\Gamma^k_{ij} = \\frac{1}{2} g^{kl} \\left(\\frac{\\partial g_{jl}}{\\partial x^i} + \\frac{\\partial g_{il}}{\\partial x^j} - \\frac{\\partial g_{ij}}{\\partial x^l}\\right)\n\n% Riemann curvature tensor\nR^l_{ijk} = \\frac{\\partial \\Gamma^l_{jk}}{\\partial x^i} - \\frac{\\partial \\Gamma^l_{ik}}{\\partial x^j} + \\Gamma^l_{im}\\Gamma^m_{jk} - \\Gamma^l_{jm}\\Gamma^m_{ik}\n\n% Geodesic equation\n\\frac{d^2 x^i}{dt^2} + \\Gamma^i_{jk} \\frac{dx^j}{dt} \\frac{dx^k}{dt} = 0"} />

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## Theorem Environments

### Setting Up Theorems

<LatexSource filename="example.tex" source={"% In preamble\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{corollary}[theorem]{Corollary}\n\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{example}[theorem]{Example}\n\n\\theoremstyle{remark}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem{note}[theorem]{Note}"} />

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### Using Theorem Environments

<LatexSource filename="example.tex" source={"\\begin{theorem}[Fermat's Last Theorem]\n\\label{thm:fermat}\nFor $n > 2$, there are no three positive integers $a$, $b$, and $c$\nthat satisfy the equation $a^n + b^n = c^n$.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is beyond the scope of this document.\nSee Wiles (1995) for details.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:helper}\nEvery finite integral domain is a field.\n\\end{lemma}\n\nBy Theorem~\\ref{thm:fermat} and Lemma~\\ref{lem:helper}, we conclude..."} />

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## Advanced Symbols and Notation

### Special Alphabets

| Style        | Command               | Example               |
| :----------- | :-------------------- | :-------------------- |
| Blackboard   | `\mathbb{R}`          | $\mathbb{R}$          |
| Calligraphic | `\mathcal{L}`         | $\mathcal{L}$         |
| Fraktur      | `\mathfrak{g}`        | $\mathfrak{g}$        |
| Script       | `\mathscr{F}`         | $\mathscr{F}$         |
| Bold         | `\mathbf{v}`          | $\mathbf{v}$          |
| Bold symbol  | `\boldsymbol{\alpha}` | $\boldsymbol{\alpha}$ |

### Advanced Operators

<LatexSource filename="example.tex" source={"% Tensor products\nV \\otimes W, \\quad \\bigotimes_{i=1}^n V_i\n\n% Direct sums\nV \\oplus W, \\quad \\bigoplus_{i=1}^n V_i\n\n% Wedge products\n\\alpha \\wedge \\beta\n\n% Cup and cap products\n\\alpha \\cup \\beta, \\quad \\alpha \\cap \\beta\n\n% Hom and End\n\\text{Hom}(V, W), \\quad \\text{End}(V)"} />

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## Tips for Advanced Mathematics

<CardGroup cols={2}>
  <Card title="Use Semantic Commands" icon="code">
    Define `\newcommand{\R}{\mathbb{R}}` for frequently used symbols
  </Card>

  <Card title="Organize Theorems" icon="list-ol">
    Use consistent numbering across theorem-like environments
  </Card>

  <Card title="Clear Notation" icon="eye">
    Define all notation clearly when first introduced
  </Card>

  <Card title="Proper Spacing" icon="ruler">
    Use `\,` and `\:` for fine-tuning mathematical spacing
  </Card>
</CardGroup>

## Further Reading

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

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

  <Card title="Physics Mathematics" icon="atom" href="/learn/latex/specialized-notation/physics">
    Physics-specific notation
  </Card>

  <Card title="Matrices Guide" icon="grid" href="/learn/latex/mathematics/matrices">
    Matrix typesetting
  </Card>
</CardGroup>