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Change of variables

In mathematics, particularly in calculus, a change of variables is a substitution technique that replaces the original variables in an integral or differential expression with new variables, often to simplify the computation by transforming the domain of integration or the form of the integrand.^[1] This method extends the basic substitution rule from single-variable calculus to multiple integrals, where it relies on the Jacobian determinant to account for the distortion of volumes or areas under the transformation.^[2] The core principle is encapsulated in the change of variables theorem, which states that for a differentiable and invertible transformation \mathbf{T}: \mathbb{R}^n \to \mathbb{R}^n with non-zero Jacobian determinant, the integral of a function f over a region D in the original variables equals the integral of f \circ \mathbf{T} times the absolute value of the Jacobian determinant over the transformed region \mathbf{T}^{-1}(D).^[2] For double integrals, this takes the form \iint_D f(x,y) \, dx \, dy = \iint_{D^*} f(g(u,v), h(u,v)) \left| \frac{\partial(x,y)}{\partial(u,v)} \right| \, du \, dv, where x = g(u,v) and y = h(u,v) define the transformation, and the Jacobian \frac{\partial(x,y)}{\partial(u,v)} = \det \begin{pmatrix} g_u & g_v \\ h_u & h_v \end{pmatrix}.^[1] A similar formula applies to triple and higher-dimensional integrals, making the technique essential for evaluating integrals in non-Cartesian coordinates, such as polar, cylindrical, or spherical systems.^[3] This approach is particularly valuable in multivariable calculus for handling complicated regions, like ellipses or paraboloids, by mapping them to simpler shapes, such as circles or rectangles, thereby reducing computational complexity.^[4] The transformation must be a diffeomorphism—smooth, bijective, and with a continuously differentiable inverse—to ensure the theorem's validity and preserve integrability.^[2] Applications extend beyond pure mathematics to physics and engineering, where change of variables facilitates solving problems in fluid dynamics, electromagnetism, and optimization by aligning coordinates with symmetries in the system.^[3]

Basics and Motivations

Simple Example

The change of variables technique, also known as substitution, simplifies the evaluation of integrals by introducing a new variable that transforms a complicated integrand into a more manageable form. Consider an integral of the form \int [f(x)](/page/F/X) \, dx; by setting u = g(x) where g is a differentiable function with a differentiable inverse, we have du = g'(x) \, dx, allowing the integral to be rewritten as \int f(g^{-1}(u)) \frac{du}{g'(g^{-1}(u))}. In practice, when the integrand matches the composition f(g(x)) g'(x), this simplifies directly to \int f(u) \, du, preserving the integral's value while facilitating computation.^[5] A straightforward example illustrates this process: evaluate the indefinite integral \int (2x + 1)^5 \, dx. Begin by choosing the substitution u = 2x + 1, which identifies the inner linear expression raised to a power. Differentiating gives du = 2 \, dx, or equivalently, dx = \frac{du}{2}. Substituting into the original integral yields \int u^5 \cdot \frac{du}{2} = \frac{1}{2} \int u^5 \, du. The antiderivative of u^5 is \frac{u^6}{6}, so \frac{1}{2} \cdot \frac{u^6}{6} + C = \frac{u^6}{12} + C. Back-substituting u = 2x + 1 produces the final result \frac{(2x + 1)^6}{12} + C. This step-by-step approach—selecting u, computing du, substituting, integrating with respect to u, and reversing the substitution—demonstrates how the method reverses the chain rule to simplify integration.^[5]^[6] Geometrically, substitution reparameterizes the integration along the real line, where the factor \frac{1}{|g'(x)|} in the rewritten integral accounts for how the transformation stretches or compresses intervals on the x-axis when mapped to the u-axis, ensuring the total "area" under the curve remains unchanged.^[5] This technique not only simplifies computation but also extends to higher dimensions via the Jacobian determinant, as explored in later sections.^[7]

Historical Context and Motivations

The concept of change of variables emerged in the late 17th century as part of the foundational work in calculus by Gottfried Wilhelm Leibniz. During his development of integral calculus between 1672 and 1676, Leibniz employed substitution techniques to simplify the evaluation of definite integrals, such as transforming the integral for the area of a unit circle's quadrant using the relation x = \frac{z^2}{1 + z^2} to yield the known value \pi/4. This algebraic substitution allowed him to convert complex expressions into more manageable forms, building on earlier geometric methods from Pascal and Cavalieri. Additionally, Leibniz applied similar substitution principles in arc length calculations, where he used differentials like ds = \sqrt{dx^2 + dy^2} to approximate curve lengths, integrating substituted forms to handle specific parametric curves without modern notation.^[8] In the 18th century, Leonhard Euler extended these ideas to multivariable settings, motivated by the need to evaluate double integrals over regions with irregular boundaries. In his 1769 paper "De formulis integralibus duplicatis," Euler introduced the change of variables formula for double integrals, demonstrating how transformations could preserve the integral's value while simplifying the domain, such as mapping rectangular regions to curved ones. This work was further generalized by Joseph-Louis Lagrange to triple integrals in the 1770s, emphasizing the method's utility in higher dimensions. Euler's contributions formalized the technique's role in calculus, shifting from ad hoc substitutions to systematic transformations.^[9]^[10] The 19th century brought rigorous formalization through the efforts of Augustin-Louis Cauchy and Bernhard Riemann, who addressed foundational issues in integration that underpinned change of variables. Cauchy's 1821 Cours d'analyse defined the definite integral via limits of sums with continuity assumptions, enabling precise substitution rules for single-variable cases by ensuring the transformation's differentiability. Riemann's 1854 habilitation thesis refined this by introducing the modern definition of the Riemann integral as a limit over partitions, which supported change of variables theorems under weaker conditions, including for absolutely continuous functions. These developments resolved earlier ambiguities in Leibnizian and Eulerian approaches, ensuring the method's validity in analysis.^[11] The primary motivations for developing change of variables stemmed from the challenges of handling composite functions in differentiation and integration, as well as adapting to natural coordinate systems in geometry and physics. In differentiation, it facilitated the chain rule for rates of change in nested variables; in integration, it transformed non-standard forms into recognizable ones, such as using polar coordinates x = r \cos \theta, y = r \sin \theta to integrate over circular domains, avoiding cumbersome Cartesian setups. This approach reduced computational complexity by aligning variables with problem symmetries, preserved invariant measures like areas or volumes under suitable transformations, and enabled exploitation of geometric properties in applications like curve rectification or surface area computations.^[9]

Formal Foundations

Single-Variable Case

In the single-variable case, a change of variables is defined using a differentiable bijection \phi: U \to V between open intervals U and V in \mathbb{R}, where the transformation x = \phi(u) maps a function f defined on V to the composite function f \circ \phi defined on U. This substitution simplifies the analysis or computation of derivatives and integrals by reparameterizing the problem in terms of the new variable u.^[6] For differentiation, the chain rule provides the key result under appropriate conditions. Suppose f: V \to \mathbb{R} is differentiable at \phi(u) and \phi: U \to V is differentiable at u \in U, with \phi continuous at u. Then the composite function h(u) = f(\phi(u)) is differentiable at u, and

h'(u) = f'(\phi(u)) \cdot \phi'(u).

The proof relies on the limit definition of the derivative. Consider

h'(u) = \lim_{\Delta u \to 0} \frac{f(\phi(u + \Delta u)) - f(\phi(u))}{\Delta u}.

Assuming \phi is differentiable, \Delta x = \phi(u + \Delta u) - \phi(u) = \phi'(u) \Delta u + o(\Delta u) as \Delta u \to 0. Substituting yields

h'(u) = \lim_{\Delta u \to 0} \left[ \frac{f(\phi(u) + \Delta x) - f(\phi(u))}{\Delta x} \cdot \frac{\Delta x}{\Delta u} \right] = f'(\phi(u)) \cdot \phi'(u),

where the first limit is f'(\phi(u)) by differentiability of f and the second is \phi'(u) by differentiability of \phi. Continuity of \phi' ensures the limits exist uniformly.^[12] For integration, the substitution theorem applies to Riemann integrals. Assume \phi: [c, d] \to [a, b] is continuous and strictly increasing with \phi(c) = a, \phi(d) = b, and \phi' exists and is integrable on [c, d]. Let f: [a, b] \to \mathbb{R} be continuous. Then

\int_a^b f(x) \, dx = \int_c^d f(\phi(u)) \phi'(u) \, du.

The proof uses the Fundamental Theorem of Calculus (FTC). Define F(t) = \int_a^t f(x) \, dx, so F'(t) = f(t) by FTC Part 2 since f is continuous. By the chain rule, (F \circ \phi)'(u) = f(\phi(u)) \phi'(u). Integrating both sides from c to d and applying FTC Part 1 gives

\int_c^d (F \circ \phi)'(u) \, du = F(\phi(d)) - F(\phi(c)) = \int_a^b f(x) \, dx,

which matches the left side. The strict monotonicity ensures \phi is a bijection and preserves the orientation of the interval.^[13] If \phi is not monotonic, the general formula incorporates an absolute value to account for sign changes in \phi':

\int_a^b f(x) \, dx = \int_c^d f(\phi(u)) |\phi'(u)| \, du,

provided f \circ \phi \cdot |\phi'| is Riemann integrable. This requires splitting the domain of integration at critical points where \phi' = 0 or changes sign, applying the monotonic case to each subinterval, and summing the results. The absolute value arises because the Riemann integral measures "signed length," but substitution preserves the total measure regardless of direction.^[14] Edge cases include constant substitutions and improper integrals. If \phi(u) = k (constant), then \phi'(u) = 0, so \int_c^d f(\phi(u)) \phi'(u) \, du = 0 for bounded continuous f, matching the original integral over a point. For improper integrals, such as \int_a^\infty f(x) \, dx where the limit exists, substitution \phi: [c, \infty) \to [a, \infty) with \phi strictly increasing, continuous, differentiable, and \phi' > 0 integrable yields \int_c^\infty f(\phi(u)) \phi'(u) \, du, evaluated as \lim_{d \to \infty} \int_c^d f(\phi(u)) \phi'(u) \, du, preserving convergence under the theorem's conditions.^[15]

Multivariable Case

In the multivariable case, a change of variables is formalized through a differentiable map \Phi: U \subset \mathbb{R}^n \to V \subset \mathbb{R}^n, where U and V are open sets, and the transformation expresses points x \in V as x = \Phi(u) for u \in U.^[2] Such a map \Phi is required to be a local diffeomorphism, meaning it is smooth (C^1), bijective onto its image, and locally invertible with a smooth inverse, ensuring the transformation preserves the structure of the space.^[2] The Jacobian matrix J_\Phi(u) of the map \Phi at a point u \in U is the n \times n matrix whose entries are the partial derivatives \frac{\partial x_i}{\partial u_j} for i, j = 1, \dots, n, representing the best linear approximation to \Phi near u.^[16] The determinant \det(J_\Phi(u)) quantifies the local scaling effect of the transformation on volumes: specifically, it measures how the map distorts infinitesimal volumes, with |\det(J_\Phi(u))| giving the absolute scaling factor for the volume of small parallelepipeds under the linear approximation J_\Phi(u).^[16] For the transformation to be valid and orientation-preserving, \det(J_\Phi(u)) > 0 at all points in U, preventing reversal of the coordinate system's handedness.^[17] A proof sketch for the volume-preserving property relies on the linear approximation: near u, \Phi behaves like the affine map x \approx \Phi(u) + J_\Phi(u)(v - u), where the image of the unit cube in the v-coordinates has volume |\det(J_\Phi(u))| times the original, as the determinant computes the signed volume of the parallelepiped spanned by the columns of J_\Phi(u).^[16] This local scaling extends to the global transformation under the diffeomorphism condition. The Jacobian matrix itself underlies the multivariable chain rule for differentiation, providing the derivative of composite functions.^[2] The inverse function theorem connects directly to these concepts: if \det(J_\Phi(u)) \neq 0, then \Phi is locally invertible near u, with the inverse's Jacobian given by \det(J_{\Phi^{-1}}(\Phi(u))) = 1 / \det(J_\Phi(u)), guaranteeing the existence of a smooth local inverse and thus the diffeomorphism property.^[2]

Differentiation Applications

Chain Rule in Single Variables

The chain rule in single variables provides a fundamental method for differentiating composite functions, where the output of one function serves as the input to another. Consider a composite function f(\phi(u)), where f and \phi are differentiable. The derivative with respect to u is given by \frac{d}{du} [f(\phi(u))] = f'(\phi(u)) \cdot \phi'(u). This rule arises from the need to account for the rate of change of the inner function \phi when computing the overall rate of change. To derive the chain rule from first principles, start with the definition of the derivative. The derivative of the composite function at u is

\lim_{h \to 0} \frac{f(\phi(u + h)) - f(\phi(u))}{h}.

Let \Delta \phi = \phi(u + h) - \phi(u), so the expression becomes

\lim_{h \to 0} \frac{f(\phi(u) + \Delta \phi) - f(\phi(u))}{h} = \lim_{h \to 0} \left( \frac{f(\phi(u) + \Delta \phi) - f(\phi(u))}{\Delta \phi} \cdot \frac{\Delta \phi}{h} \right).

Assuming \phi is differentiable, \lim_{h \to 0} \frac{\Delta \phi}{h} = \phi'(u). Similarly, since f is differentiable at \phi(u), \lim_{\Delta \phi \to 0} \frac{f(\phi(u) + \Delta \phi) - f(\phi(u))}{\Delta \phi} = f'(\phi(u)), and \Delta \phi \to 0 as h \to 0. Thus, the limit simplifies to f'(\phi(u)) \cdot \phi'(u). This derivation relies on the continuity and differentiability assumptions of f and \phi. A classic example is differentiating y = \sin(x^2). Here, let \phi(x) = x^2 and f(u) = \sin u, so \frac{dy}{dx} = \cos(x^2) \cdot 2x. This illustrates how the chain rule multiplies the derivative of the outer function by that of the inner one. Another application appears in parametric curves, such as position x(t) and velocity v = \frac{dx}{dt}. If x = \phi(u) and u = u(t), then v = \frac{dx}{dt} = \phi'(u) \cdot \frac{du}{dt}, enabling computation of rates in terms of intermediate variables. Implicit differentiation employs the chain rule as a change-of-variables technique when an equation defines y implicitly as a function of x, without solving explicitly. Differentiate both sides with respect to x, treating y as a function of x, so terms involving y require the chain rule: \frac{dy}{dx} multiplies the derivative with respect to y. For instance, in x^2 + y^2 = 1, differentiating yields $2x + 2y \frac{dy}{dx} = 0, so \frac{dy}{dx} = -\frac{x}{y}. This method is essential for relations not expressible as y = f(x).^[18] For higher-order derivatives of composite functions, the chain rule extends via Faà di Bruno's formula, which generalizes to the nth derivative. For the second derivative,

\frac{d^2}{du^2} [f(\phi(u))] = f''(\phi(u)) [\phi'(u)]^2 + f'(\phi(u)) \phi''(u).

This formula, first published by Francesco Faà di Bruno in 1855, accounts for all ways the inner function's derivatives contribute to the outer one's higher derivatives through Bell partitions. It is particularly useful in analyzing curvature or acceleration in parametric forms. Computational tips for applying the chain rule include identifying the outermost function first and working inward, as in the \sin(x^2) example. For products or quotients raised to powers, such as y = \frac{x^2 (x+1)^3}{e^x}, logarithmic differentiation simplifies the process: take \ln y = \ln(x^2) + 3\ln(x+1) - x, differentiate to get \frac{1}{y} \frac{dy}{dx} = \frac{2}{x} + \frac{3}{x+1} - [1](/page/1), then multiply by y. This leverages the chain rule on the logarithm to avoid repeated product/quotient rules.^[19]

Jacobian and Multivariable Differentiation

In multivariable calculus, the chain rule extends to compositions of vector-valued functions, where the Jacobian matrix plays a central role. Consider functions F: \mathbb{R}^m \to \mathbb{R}^n and G: \mathbb{R}^k \to \mathbb{R}^m, both differentiable at the appropriate points. The Jacobian matrix of the composition F \circ G at a point \mathbf{t} \in \mathbb{R}^k is given by D(F \circ G)(\mathbf{t}) = DF(G(\mathbf{t})) \cdot DG(\mathbf{t}), where DF and DG denote the Jacobian matrices of F and G, respectively. This matrix equation generalizes the single-variable chain rule (f \circ g)' = f'(g) g' to higher dimensions, representing the linear approximation of the composite map. The total derivative under a change of variables \mathbf{x} = \Phi(\mathbf{u}), where \Phi: \mathbb{R}^k \to \mathbb{R}^m is differentiable, is captured by d\mathbf{x} = J(\Phi)(\mathbf{u}) \, d\mathbf{u}, with J(\Phi) the Jacobian matrix of \Phi. This relation provides the first-order linear approximation of the transformation near a point, essential for understanding how infinitesimal changes in the new variables \mathbf{u} map to changes in the original variables \mathbf{x}. For instance, in computing partial derivatives after a variable change, the chain rule yields \frac{\partial f}{\partial u_i} = \sum_j \frac{\partial f}{\partial x_j} \frac{\partial x_j}{\partial u_i}, or in vector form, \nabla_{\mathbf{u}} (f \circ \Phi) = J(\Phi)^T [\nabla_{\mathbf{x}} f](/page/Gradient). Equivalently, the gradient in the original coordinates transforms as \nabla_{\mathbf{x}} f = J(\Phi)^{-T} \nabla_{\mathbf{u}} (f \circ \Phi), assuming J(\Phi) is invertible. This framework is particularly useful for computing gradients in curvilinear coordinate systems, where the Jacobian facilitates the inclusion of scale factors. In orthogonal curvilinear coordinates, the gradient operator takes the form \nabla f = \sum_\alpha \frac{1}{h_\alpha} \frac{\partial f}{\partial y_\alpha} \hat{e}_\alpha, with scale factors h_\alpha = \left\| \frac{\partial \mathbf{r}}{\partial y_\alpha} \right\| derived from the Jacobian entries of the position vector \mathbf{r}(y_1, \dots, y_k). These scale factors account for the stretching or compression in each coordinate direction, enabling efficient evaluation of directional derivatives without reverting to Cartesian components. For higher-order derivatives, the Hessian matrix under a change of variables is more complex, involving both the Jacobian of the transformation and second derivatives of the coordinate map. Specifically, for a scalar function f, the Hessian of f \circ \Phi at \mathbf{u} is H_{\mathbf{u}}(f \circ \Phi) = J(\Phi)^T H_{\mathbf{x}} f \, J(\Phi) + \sum_k (\nabla_{\mathbf{x}} f)_k \, D^2 \Phi_k, where D^2 \Phi_k is the Hessian of the k-th component of \Phi. This formula arises from applying the multivariable chain rule twice, highlighting the quadratic approximation's sensitivity to the curvature of the transformation.^[20]

Integration Applications

Substitution in Single Integrals

Substitution in single integrals, also known as u-substitution, applies the change of variables theorem to one-dimensional cases, transforming ∫ f(x) dx into ∫ f(g^{-1}(u)) |du / g'(g^{-1}(u))| du where u = g(x) and g is differentiable with g' ≠ 0 on the interval of integration.^[21] This method simplifies integrands by reversing the chain rule, particularly useful for composites like ∫ f(g(x)) g'(x) dx = ∫ f(u) du. For definite integrals, the substitution requires adjusting the limits of integration according to the new variable u, ensuring the mapping preserves the orientation or accounting for reversals via the absolute value of the derivative.^[14] A classic example is the trigonometric substitution for integrals involving √(1 - x²). Consider the indefinite integral ∫ dx / √(1 - x²). Let x = sin θ, so dx = cos θ dθ and √(1 - x²) = cos θ (assuming θ in [-π/2, π/2] where cos θ ≥ 0). The integral becomes ∫ (cos θ dθ) / cos θ = ∫ dθ = θ + C = arcsin x + C.^[22] For the definite integral from 0 to 1, the limits change from x = 0 (θ = 0) to x = 1 (θ = π/2), yielding ∫_0^{π/2} dθ = π/2, which matches the area of a quarter unit circle.^[23] Another example illustrates bound adjustments and sign handling in definite integrals. Evaluate ∫_0^1 x √(1 - x) dx. Let u = 1 - x, so du = -dx and x = 1 - u. The limits shift from x = 0 (u = 1) to x = 1 (u = 0), transforming the integral to -∫_1^0 (1 - u) √u du = ∫_0^1 (1 - u) u^{1/2} du. Expanding gives ∫_0^1 (u^{1/2} - u^{3/2}) du = [ (2/3) u^{3/2} - (2/5) u^{5/2} ]_0^1 = 2/3 - 2/5 = 4/15. The negative sign from du flips the bounds, equivalent to taking |du/dx| = 1 to preserve the positive measure.^[14] Common techniques extend substitution for specific forms. Euler substitutions simplify integrals with radicals of linear or quadratic arguments, such as ∫ R(x, √(ax² + bx + c)) dx where R is rational. The first Euler substitution sets √(ax² + bx + c) = t + √a x (for a > 0), rationalizing the radical into a quadratic in t; the second uses √(ax² + bx + c) = t x for b² - 4ac > 0; the third applies √(ax² + bx + c) = t (a x + b/2) for b² - 4ac < 0. These yield rational functions integrable via partial fractions.^[24] Hyperbolic substitutions handle forms like √(x² - 1) or √(x² + 1), analogous to trigonometric but using identities like cosh² u - sinh² u = 1. For ∫ dx / √(x² - 1), let x = cosh u (u ≥ 0), dx = sinh u du, simplifying to ∫ du = arcosh x + C, useful for exponential-related integrands.^[25] Substitution can fail if the mapping g(x) = u is not bijective over the interval, as the change of variables assumes a one-to-one correspondence to avoid over- or under-counting contributions. For non-monotonic g, such as g(x) = x² on [-1, 1], the integral must be split into piecewise monotonic subintervals (e.g., [-1, 0] and [0, 1]) where the substitution applies separately, then summed. Failure to do so distorts the result, as the inverse is multi-valued. Numerically, the factor |du/dx| in the substitution accounts for the "speed" of the mapping, scaling the infinitesimal dx to du by the local stretching or compression rate; a large |du/dx| compresses intervals in x to smaller ones in u, concentrating the integral's contribution, while a small value expands them. This ensures the transformed integral preserves the original's value under the Lebesgue measure for improper cases.^[14]

Change of Variables in Multiple Integrals

The change of variables theorem for multiple integrals provides a method to transform the variables of integration in Riemann integrals over regions in \mathbb{R}^n, facilitating the evaluation of integrals that are difficult in the original coordinates. This theorem generalizes the substitution rule from single-variable calculus to higher dimensions, incorporating the Jacobian determinant to account for the scaling of volume elements under the transformation. For a continuously differentiable bijection \Phi: U \to V between open sets in \mathbb{R}^n, where D \subset V is a bounded region and f: D \to \mathbb{R} is a bounded continuous function, the theorem states that

\int_D f(\mathbf{x}) \, d\mathbf{x} = \int_{\Phi^{-1}(D)} f(\Phi(\mathbf{u})) \left| \det J_\Phi(\mathbf{u}) \right| \, d\mathbf{u},

where J_\Phi(\mathbf{u}) is the Jacobian matrix of \Phi at \mathbf{u}, and \det J_\Phi(\mathbf{u}) is its determinant.^[2]^[1] This formula applies to Riemann integrals and assumes \Phi is a diffeomorphism, ensuring the transformation is invertible with a continuously invertible Jacobian.^[2] The proof of the theorem relies on approximating the integral via Riemann sums over partitions of the domain, where the transformation distorts infinitesimal volume elements. Specifically, for a small parallelepiped in the \mathbf{u}-space with volume |\Delta \mathbf{u}|, its image under \Phi approximates a parallelepiped in the \mathbf{x}-space with volume |\det J_\Phi(\mathbf{u})| \cdot |\Delta \mathbf{u}|, as the Jacobian matrix linearly maps the edges and scales volumes by its determinant's absolute value. Summing these contributions and taking the limit as the partition refines yields the integral transformation, with the absolute value ensuring the volume measure remains positive regardless of orientation.^[2] This approach highlights the Jacobian's role in preserving the integral's value through local linear approximations of the mapping.^[26] A classic application arises when evaluating \iint_D x \, dA over the disk D = \{(x,y) \mid x^2 + y^2 \leq R^2\} using polar coordinates, where the transformation is x = r \cos \theta, y = r \sin \theta, with $0 \leq r \leq R and $0 \leq \theta \leq 2\pi. The Jacobian matrix is

J = \begin{pmatrix} \cos \theta & -r \sin \theta \\ \sin \theta & r \cos \theta \end{pmatrix},

and \det J = r, so |\det J| = r since r \geq 0. Substituting yields

\iint_D x \, dA = \int_0^{2\pi} \int_0^R (r \cos \theta) \cdot r \, dr \, d\theta = \int_0^{2\pi} \cos \theta \, d\theta \int_0^R r^2 \, dr = 0,

as the angular integral vanishes by symmetry, demonstrating the theorem's utility in simplifying symmetric regions.^[1] The absolute value in the formula addresses orientation: if \det J_\Phi > 0, the transformation preserves orientation (right-handed to right-handed), while \det J_\Phi < 0 reverses it, but the absolute value ensures the unsigned Riemann integral remains positive and correctly scaled for volume. In contexts involving differential forms or oriented integrals, the signed determinant is used instead to reflect the orientation change.^[2]^[26] The theorem is compatible with Fubini's theorem, allowing the transformed multiple integral to be evaluated as an iterated integral over rectangular or simple regions in the new variables, provided the integrand satisfies the necessary continuity conditions for Fubini to apply post-transformation.^[27]^[1] For a generalization to Lebesgue integrals, the theorem extends under milder measurability assumptions, replacing Riemann sums with measure-theoretic limits.

Coordinate Transformations

Polar and Cylindrical Coordinates

In polar coordinates, the change of variables from Cartesian coordinates (x, y) to (r, \theta) is given by x = r \cos \theta and y = r \sin \theta, where r \geq 0 and \theta \in [0, 2\pi).^[1] The Jacobian determinant for this transformation is r, so the area element transforms as dx\, dy = r\, dr\, d\theta.^[28] This factor arises from the partial derivatives of the transformation and is essential for integrals over regions with circular symmetry.^[29] A classic application is evaluating the Gaussian integral \iint_{-\infty}^{\infty} e^{-(x^2 + y^2)} \, dx\, dy. Switching to polar coordinates yields \int_0^{2\pi} \int_0^{\infty} r e^{-r^2} \, dr\, d\theta = \pi, simplifying the computation by exploiting radial symmetry.^[30] For differentiation, the gradient of a scalar function f(r, \theta) in polar coordinates is \nabla f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r} \frac{\partial f}{\partial \theta} \hat{\theta}, reflecting the scaled angular component due to the coordinate geometry.^[31] The Laplacian operator transforms to \Delta f = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2}, which is derived via the chain rule and facilitates solving partial differential equations in polar settings.^[32] Cylindrical coordinates extend polar coordinates to three dimensions by keeping z unchanged, so x = r \cos \theta, y = r \sin \theta, z = z, with r \geq 0, \theta \in [0, 2\pi), and z \in \mathbb{R}.^[33] The Jacobian determinant remains r, transforming the volume element to dV = r\, dr\, d\theta\, dz, suitable for regions with axial symmetry along the z-axis.^[16] For example, the area of an annulus between radii a and b (with $0 < a < b) is computed as \int_0^{2\pi} \int_a^b r\, dr\, d\theta = \pi (b^2 - a^2), directly using the Jacobian for the radial integration.^[34] Line integrals around circular paths, such as \int_C \mathbf{F} \cdot d\mathbf{r} for a vector field in the plane, simplify in polar form by parameterizing \mathbf{r}( \theta ) = (r \cos \theta, r \sin \theta) with d\mathbf{r} = r (-\sin \theta, \cos \theta) d\theta, yielding \int_0^{2\pi} \mathbf{F}(r \cos \theta, r \sin \theta) \cdot (-r \sin \theta, r \cos \theta) \, d\theta.^[1] These transformations have limitations: the Jacobian vanishes at r = 0, introducing a singularity that requires careful handling in integrals or derivatives near the origin, and \theta's periodicity demands consistent branch choices to avoid discontinuities.^[35]

Spherical and Other Curvilinear Systems

Spherical coordinates provide a natural change of variables for problems exhibiting spherical symmetry in three-dimensional space, transforming from Cartesian coordinates (x, y, z) to radial distance \rho, polar angle \phi, and azimuthal angle \theta. The transformation is given by

\begin{align*} x &= \rho \sin \phi \cos \theta, \\ y &= \rho \sin \phi \sin \theta, \\ z &= \rho \cos \phi, \end{align*}

where \rho \geq 0, $0 \leq \phi \leq \pi, and $0 \leq \theta < 2\pi.^[1] The Jacobian determinant for this transformation, essential for changing variables in integrals, is \det J = \rho^2 \sin \phi.^[1] This factor accounts for the distortion in volume elements under the coordinate change.^[36] In integration, the volume element in spherical coordinates becomes dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta.^[1] For example, the volume of the unit ball \rho \leq 1 is computed as

\iiint 1 \, dV = \int_0^\pi \int_0^{2\pi} \int_0^1 \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi = \frac{4\pi}{3},

demonstrating the utility of the Jacobian in simplifying spherical integrals.^[1] Such transformations are particularly effective for integrating over spherically symmetric regions, like spheres or balls.^[36] For differentiation, the gradient of a scalar function f in spherical coordinates is

\nabla f = \frac{\partial f}{\partial \rho} \hat{\rho} + \frac{1}{\rho} \frac{\partial f}{\partial \phi} \hat{\phi} + \frac{1}{\rho \sin \phi} \frac{\partial f}{\partial \theta} \hat{\theta}.

^[37] The divergence of a vector field \mathbf{F} = F_\rho \hat{\rho} + F_\phi \hat{\phi} + F_\theta \hat{\theta} is

\nabla \cdot \mathbf{F} = \frac{1}{\rho^2} \frac{\partial}{\partial \rho} (\rho^2 F_\rho) + \frac{1}{\rho \sin \phi} \frac{\partial}{\partial \phi} (\sin \phi F_\phi) + \frac{1}{\rho \sin \phi} \frac{\partial F_\theta}{\partial \theta},

and the curl is

\nabla \times \mathbf{F} = \frac{1}{\rho \sin \phi} \left[ \frac{\partial}{\partial \phi} (\sin \phi F_\theta) - \frac{\partial F_\phi}{\partial \theta} \right] \hat{\rho} + \frac{1}{\rho} \left[ \frac{1}{\sin \phi} \frac{\partial F_\rho}{\partial \theta} - \frac{\partial}{\partial \rho} (\rho F_\theta) \right] \hat{\phi} + \frac{1}{\rho} \left[ \frac{\partial}{\partial \rho} (\rho F_\phi) - \frac{\partial F_\rho}{\partial \phi} \right] \hat{\theta}.

^[38] These expressions arise from the general formulas for orthogonal curvilinear coordinates, adapted to the scale factors in spherical systems: h_\rho = 1, h_\phi = \rho, h_\theta = \rho \sin \phi.^[38] Other curvilinear systems include toroidal coordinates, suitable for ring-like or toroidal regions such as those in plasma physics or vortex flows. The transformation from Cartesian to toroidal coordinates (\xi, \eta, \phi) is

\begin{align*} x &= \frac{a \sinh \eta \cos \phi}{\cosh \eta - \cos \xi}, \\ y &= \frac{a \sinh \eta \sin \phi}{\cosh \eta - \cos \xi}, \\ z &= \frac{a \sin \xi}{\cosh \eta - \cos \xi}, \end{align*}

with $0 \leq \xi < 2\pi, \eta \geq 0, $0 \leq \phi < 2\pi, and a > 0 a scale parameter.^[39] The scale factors are h_\xi = h_\eta = a / (\cosh \eta - \cos \xi) and h_\phi = a \sinh \eta / (\cosh \eta - \cos \xi), yielding a Jacobian determinant of |\det J| = a^3 \sinh \eta / (\cosh \eta - \cos \xi)^3.^[39] This system facilitates integration over toroidal volumes by aligning coordinates with the geometry.^[39] Applications of these transformations abound in physics, particularly for computing gravitational potentials around spherically symmetric masses, where the change to spherical coordinates simplifies the Poisson equation \nabla^2 \Phi = 4\pi [G](/page/G) \rho due to radial symmetry.^[40] For instance, the potential outside a uniform sphere integrates straightforwardly using the spherical volume element, yielding \Phi(r) = -GM/r for r greater than the sphere's radius.^[40]

Advanced and Specialized Uses

In Differential Equations

Change of variables is a fundamental technique in the solution of differential equations, allowing the transformation of complex equations into simpler forms that are more amenable to standard solution methods. In ordinary differential equations (ODEs), substitutions exploit the structure of the equation to reduce its order or linearity, while in partial differential equations (PDEs), they often align the equation with characteristic curves or symmetry properties to yield explicit solutions. This approach preserves the essential dynamics while simplifying computations, and it underpins many analytical methods in applied mathematics.^[41] For first-order ODEs, change of variables is particularly useful for homogeneous equations of the form \frac{dy}{dx} = f\left(\frac{y}{x}\right), where the right-hand side depends only on the ratio y/x. The substitution v = y/x, or equivalently y = v x, transforms the equation by differentiating to obtain \frac{dy}{dx} = v + x \frac{dv}{dx}, yielding

x \frac{dv}{dx} = f(v) - v

after substitution. This separates variables, allowing integration as

\int \frac{dv}{f(v) - v} = \int \frac{dx}{x}.

The result is a separable equation solvable by direct integration, demonstrating how the substitution exploits the scaling invariance of the homogeneous form.^[41]^[42] Another key application in ODEs is the Bernoulli equation, \frac{dy}{dx} + P(x) y = Q(x) y^n with n \neq 0, 1. The substitution v = y^{1-n} linearizes the nonlinearity: differentiating gives \frac{dv}{dx} = (1-n) y^{-n} \frac{dy}{dx}, so multiplying the original equation by (1-n) y^{-n} yields the linear form

\frac{dv}{dx} + (1-n) P(x) v = (1-n) Q(x).

This first-order linear ODE in v can then be solved using an integrating factor, after which back-substitution recovers y. For exact equations, substitutions may also facilitate finding an integrating factor when the equation is not immediately exact, though this is case-specific.^[43]^[44] In PDEs, the method of characteristics employs change of variables to solve first-order quasilinear equations like a(x,t,u) u_x + b(x,t,u) u_t = c(x,t,u). The characteristics are curves parameterized by \frac{dx}{ds} = a, \frac{dt}{ds} = b, \frac{du}{ds} = c, and new variables \xi = x - c t (for the transport equation u_t + c u_x = 0) align the PDE with these curves. Substituting yields u_\xi = 0 along characteristics, implying u is constant on them, so the general solution is u(x,t) = f(x - c t) for arbitrary f. This reduces the PDE to an ODE system along the characteristics.^[45]^[46] Similarity solutions arise in nonlinear PDEs with scaling symmetries, such as the heat equation u_t = u_{xx}. The change of variables \eta = x / \sqrt{t} and u(x,t) = t^{-1/2} f(\eta) (or similar scaling) transforms the PDE into an ODE in \eta:

f'' + \frac{\eta}{2} f' + \frac{1}{2} f = 0,

whose solutions capture self-similar profiles invariant under time and space rescaling. This method reveals fundamental behaviors like diffusion fronts without solving the full initial-value problem.^[47] More advanced changes of variables stem from Lie group symmetries, where infinitesimal transformations generated by a Lie algebra leave the equation invariant. For an ODE or PDE, symmetries yield invariants that define new coordinates, reducing the equation's order or dimensionality; for instance, scaling symmetries in the heat equation lead directly to the similarity variable \eta. This framework, developed by Sophus Lie, systematically identifies substitutions based on the equation's symmetry group, enabling solutions via canonical forms.^[48]^[49]

In Physics and Mechanics

In physics and mechanics, the change of variables is essential for reformulating the equations of motion in terms of generalized coordinates, which simplify the description of complex systems by exploiting symmetries and constraints. In Lagrangian mechanics, the Lagrangian function L(\mathbf{q}, \dot{\mathbf{q}}), where \mathbf{q} = (q_1, \dots, q_n) are the generalized coordinates and \dot{\mathbf{q}} their time derivatives, encodes the system's kinetic energy minus potential energy.^[50] These coordinates can be any set of independent parameters that uniquely specify the system's configuration, such as angles or lengths, rather than Cartesian positions.^[51] The equations of motion arise from the Euler-Lagrange equations: \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0 for each i. Changes of variables to new generalized coordinates Q_j = Q_j(\mathbf{q}, t) via time-dependent point transformations preserve the form of these equations, allowing the Lagrangian to be expressed in the new variables as L'(\mathbf{Q}, \dot{\mathbf{Q}}) = L(\mathbf{q}(\mathbf{Q}, t), \dot{\mathbf{q}}(\mathbf{Q}, \dot{\mathbf{Q}}, t)), up to a total time derivative that does not affect the dynamics.^[52] This invariance facilitates the analysis of systems with rotational or other symmetries. A key application is in central force problems, where polar coordinates (r, \theta) serve as generalized coordinates for a particle under a radial potential V(r). The Lagrangian becomes L = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) - V(r), leading to the Euler-Lagrange equation for \theta that implies conservation of angular momentum: p_\theta = m r^2 \dot{\theta} = \constant./04%3A_Hamilton's_Principle_and_Noether's_Theorem/4.09%3A_Example_2-__Lagrangian_Formulation_of_the_Central_Force_Problem) This conserved quantity arises from the cyclic nature of \theta in the Lagrangian, highlighting how coordinate choices reveal physical invariants.^[53] The canonical momentum p_i = \frac{\partial L}{\partial \dot{q}_i} generally differs from the linear momentum m \dot{\mathbf{x}} in non-Cartesian coordinates; for instance, in the polar example, p_\theta = m r^2 \dot{\theta} represents angular rather than linear momentum, while p_r = m \dot{r} aligns with the radial component./07%3A_Symmetries_Invariance_and_the_Hamiltonian/7.02%3A_Generalized_Momentum) This distinction is crucial in curvilinear frames, where velocities \dot{\mathbf{q}} do not directly correspond to physical velocities.^[52] In fluid mechanics, change of variables via scaling introduces dimensionless forms of the Navier-Stokes equations, such as rescaling position x' = x/L, time t' = t U/L, and velocity \mathbf{u}' = \mathbf{u}/U, where L is a characteristic length and U a velocity scale. This yields the Reynolds number \mathrm{Re} = U L / \nu as a coefficient governing the balance between inertial and viscous terms, enabling analysis of flow regimes without dimensional constants.^[54]^[55] In the Hamiltonian formulation, changes of variables in phase space (\mathbf{q}, \mathbf{p}) to new coordinates (\mathbf{Q}, \mathbf{P}) must be canonical transformations to preserve the symplectic structure, ensuring Hamilton's equations \dot{q}_i = \frac{\partial H}{\partial p_i}, \dot{p}_i = -\frac{\partial H}{\partial q_i} retain their form for the transformed Hamiltonian H'(\mathbf{Q}, \mathbf{P}).^[56] Such transformations, generated by functions like F(\mathbf{q}, \mathbf{Q}), maintain the Poisson brackets \{q_i, p_j\} = \delta_{ij} and thus the underlying geometry of phase space.^[57]

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