{
  "id": "jacobian",
  "version": "602d0dd1202a",
  "status": "published",
  "name": "Jacobian Calculator",
  "question": "What is the Jacobian?",
  "summary": "Finds the Jacobian matrix and determinant of a change of variables, checked numerically.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/jacobian-calculator",
  "markdown": "https://www.acalculator.org/math/jacobian-calculator.md",
  "kind": "function",
  "method": "Partial derivatives from a CAS, each checked; the determinant by cofactors, simplified.",
  "assumptions": [
    "Radians. An answer that fails its check is not shown."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "f": {
        "title": "Functions",
        "description": "x, y (and z) in terms of the variables, separated by commas.",
        "type": "string",
        "maxLength": 240
      },
      "v": {
        "title": "Variables",
        "description": "The variables, in order, such as u, v.",
        "type": "string",
        "maxLength": 20
      }
    }
  },
  "outputs": {
    "det": {
      "label": "Jacobian determinant",
      "description": "The determinant of the Jacobian matrix.",
      "format": "math"
    },
    "matrix": {
      "label": "Jacobian matrix",
      "description": "Row i holds the partial derivatives of function i; rows are separated by semicolons.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "u^2 - v^2, u v",
      "v": "u, v"
    },
    "outputs": {
      "det": "2u^2 + 2v^2",
      "matrix": "[2u, -2v; v, u]"
    },
    "text": "The Jacobian determinant of u^2 - v^2, u v with respect to u, v is 2u^2 + 2v^2."
  },
  "examples": [
    {
      "given": {
        "f": "u^2 - v^2, u v",
        "v": "u, v"
      },
      "expect": {
        "det": "2u^2 + 2v^2",
        "matrix": "[2u, -2v; v, u]"
      },
      "source": "OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 5.7, Ex. 5.68"
    },
    {
      "given": {
        "f": "r cos(t), r sin(t)",
        "v": "r, t"
      },
      "expect": {
        "det": "r"
      },
      "source": "OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 5.7, Ex. 5.67"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 3, section 5.7 Change of Variables in Multiple Integrals: https://openstax.org/books/calculus-volume-3/pages/5-7-change-of-variables-in-multiple-integrals"
  ],
  "related": [
    "partial-derivative",
    "double-integral",
    "triple-integral",
    "determinant"
  ],
  "changelog": []
}