Multivariable calculus

Multivariable calculus extends differentiation and integration to functions of several variables.

If $f:U\to {\mathbb{R}}^m$ is continuously differentiable and the segment from $x$ to $y$ lies in $U$, then

A partial derivative differentiates a multivariable function with respect to one variable while holding the others fixed.

A mixed partial derivative takes partial derivatives in different variables, for example $f_{xy}={\partial }_y({\partial }_{x}f)$. If both second partial derivatives are continuous in a neighbourhood, their order can be interchanged: $f_{xy}=f_{yx}$. The chain rule transforms mixed partial derivatives under a linear change of variables by composing the corresponding first-order differential operators.

A time derivative is the partial derivative of an evolving field with respect to time while holding its spatial coordinates fixed. For a quantity depending only on time, it is an ordinary derivative.

A spatial derivative is a partial derivative with respect to a spatial coordinate.

The directional derivative of $f$ at $x$ in the direction $v$ isFor a differentiable scalar function on Euclidean space, it equals $v\cdot \nabla f(x)$.

The gradient of a scalar-valued differentiable function is the vector of its first partial derivatives.

A gradient field is a vector field of the form $\nabla f$ for a scalar potential $f$. Its curl vanishes wherever the required second derivatives commute.

For a differentiable scalar function of several variables, the total differential is

The Hessian is the symmetric matrix of second partial derivatives; definiteness classifies nondegenerate critical points.

For $f(x,y)=g(x)+h(y)$, a degenerate Hessian matrix can be supplemented by the first nonzero term of each one-variable displacement. Two negative leading even powers give a strict local maximum, two positive ones a strict local minimum, and opposite signs give a saddle. Looking along the coordinate directions and bounding the separated terms avoids an inconclusive quadratic test.

A symmetric matrix $A$ is negative semidefinite when $v^{T}Av\le 0$ for every vector $v$. Equivalently, all its eigenvalues are nonpositive.

At an interior local maximum of a twice differentiable function, its Hessian matrix is negative semidefinite; at a local minimum it is positive semidefinite. For a scalar function on Euclidean space, the Laplacian is therefore nonpositive at a local maximum and nonnegative at a local minimum.

A critical point of a twice differentiable function is nondegenerate when its Hessian matrix is invertible there. A positive-definite Hessian gives a strict local minimum, a negative-definite Hessian gives a strict local maximum, and an indefinite Hessian gives a saddle point.

Vector calculus studies differentiation and integration of scalar and vector fields.

For continuously differentiable vector fields, the gradient of their dot product expands as displayed. The contraction of two Levi-Civita symbols gives $[F\times (\nabla \times G){]}_i=F_j{\partial }_i{G}_j-F_j{\partial }_j{G}_i$; adding the directional derivative cancels the second term. Interchanging the fields and using the product rule proves the identity. It does not require either field to be irrotational.

The Helmholtz decomposition separates a vector field into a gradient field and a field with zero divergence. The precise function spaces, boundary conditions and decay hypotheses determine existence and uniqueness. In the square-integrable bounded-domain formulation, the solenoidal component is selected by the Leray-Helmholtz projection.

For differentiable vector fields $A$ and $B$,

A vector field assigns a vector to every point of its domain.

A smooth family $Y_t$ assigns a tangent vector at each space-time point. Its flow maps solve ${\partial }_t{\rho }_t=Y_t\circ {\rho }_t$, with ${\rho }_0$ the identity. For a varying differential form, differentiation gives ${\partial }_t({\rho }_t^{\ast }{\alpha }_t)={\rho }_t^{\ast }({\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t)$.

A smooth time-dependent vector field on a compact manifold without boundary has flow maps throughout each compact time interval on which the field is defined. Finite-time escape is impossible on the compact state space. Uniqueness and the backward-time equation supply smooth inverses, so the flow is a smooth isotopy.

Differentiate the displayed homogeneity relation in $\lambda$ at $1$. On every regular point of $H=0$, the radial vector field $x$ is therefore perpendicular to the normal gradient and tangent to the level surface. Its flow $x(t)=e^{t}x(0)$ preserves that zero set. At a singular vertex this flow formulation remains meaningful even if there is no tangent plane.

For $r=|x|>0$, the gradient satisfies ${\partial }_{i}r=x_i/r$. The product rule and Einstein notation then give the displayed divergence formula in dimension $n$. In dimension three, its derivative matrix is symmetric, so the curl vanishes. Zero divergence forces $f(r)=C{r}^{-n}$ on each positive radial interval. This formula does not assert regularity at the origin.

The vector field $\mathbf{G}=(-y,x,0)/(x^2+y^2)$ has zero curl away from its excluded axis, but has circulation $2\pi$ around a positively oriented circle enclosing that axis. Stokes theorem gives zero total circulation for a smooth annular surface avoiding the axis, whose inner and outer boundary circles have opposite orientations. It cannot be applied directly to a disk meeting the axis, because the field is undefined there.

An integral curve follows a vector field by solving the displayed ordinary differential equation. For a smooth vector field, integral curves through initial points assemble into its local flow.

For a continuously differentiable nonzero vector field $F$, the curvature of a space curve following its direction isTake the unit tangent vector $t=\pm F/|F|$ and differentiate with respect to arc length. The derivative of $|F{|}^{-1}$ is parallel to $F$ and disappears from $t\times t^{\prime }$. Since $t\cdot t^{\prime }=0$, we have $|t\times t^{\prime }|=|t^{\prime }|=\kappa$, proving the formula. It remains valid with zero curvature, even where the usual Frenet frame is undefined.

An axisymmetric vector field is invariant under rotations about a chosen axis. In cylindrical coordinates, its components are independent of the azimuthal angle.

A solenoidal vector field has zero divergence. Magnetic fields are solenoidal because $\nabla \cdot \mathbf{B}=0$.

A scalar function on Euclidean space is spherically symmetric when it depends only on the distance $r=|x|$ from a fixed center.

If a helicity density evolves as a sum of directional derivatives along divergence-free fields tangent to the boundary, its volume integral is conserved because both terms become vanishing boundary fluxes.

For $F=(F_1,F_2,F_3)$ in Cartesian coordinates,

For the Levi-Civita connection of a Riemannian metric, the divergence of $X$ is ${\mathrm{div}}_{g}X=\mathrm{tr}(Y\mapsto {\nabla }_{Y}X)$. In coordinates it equals $(\det g{)}^{-1/2}{\partial }_i((\det g{)}^{1/2}{X}^i)$. On an oriented manifold it satisfies ${\mathcal{L}}_X{\omega }_g=({\mathrm{div}}_{g}X){\omega }_g$. The codifferential on a one-form satisfies $\delta \theta =-{\mathrm{div}}_g({\theta }^{♯})$, where $♯$ is the musical isomorphism; this follows by integration by parts or the Hodge star formula.

For a vector field directed radially from a fixed axis and depending only on distance $\rho$ from that axis, cylindrical coordinates give $\nabla \cdot [f(\rho ){\mathbf{e}}_{\rho }]=f\prime +f/\rho$. Zero divergence gives $f=A/\rho$ on any positive radial interval. Unless $A=0$, this field is singular on the axis, so applications of the divergence theorem must use a domain avoiding it or account separately for that singularity.

For a locally integrable vector field, this formula defines its divergence as a distribution. For a piecewise smooth field, a jump of its normal component across an interface adds a surface measure to the smooth divergences. A continuous normal component eliminates that measure, as follows by integration by parts on the two sides.

For $\mathbf{u}=u_r{\mathbf{e}}_r+u_{\theta }{\mathbf{e}}_{\theta }$,

For a scalar field $\varphi$ and vector field $F$,

A total divergence has the form $\nabla \cdot \mathbf{F}$. The divergence theorem converts its volume integral into a boundary flux, which vanishes when the field has suitable decay or boundary conditions.

For a twice continuously differentiable vector field, commuting partial derivatives and antisymmetry of the Levi-Civita symbol give

The curl of $F$ has components

A continuously differentiable vector field on an open subset of three-dimensional Euclidean space is irrotational when its curl is zero. Under the Euclidean identification $\alpha =X_{i}d{x}^i$, this says that the differential one-form $\alpha$ is a closed differential form. The Poincare lemma gives a potential of a conservative vector field locally. A global potential additionally requires zero line integrals around every closed curve; simply connected domains ensures this. On the punctured plane extended in the third direction, $X=(-y,x,0)/(x^2+y^2)$ has zero curl but integral $2\pi$ around a circle, so it is locally conservative and has no single-valued global potential.

A vector potential for a divergence-free vector field $F$ is a vector field $A$ satisfying $F=\nabla \times A$. It is not unique because adding a gradient leaves its curl unchanged.

Let $B$ be a continuously differentiable solenoidal vector field on an open star-shaped set containing the origin. Thenis a vector potential for $B$. For $D={\int }_0^{1}tB(tx)dt$, differentiation gives $\nabla \cdot D=0$ and $\nabla \times (D\times x)=2D+(x\cdot \nabla )D={\int }_0^1{\partial }_t(t^{2}B(tx))dt=B(x)$. Boundedness near the origin removes the lower endpoint. A field defined only on a punctured domain need not satisfy these hypotheses.

For a twice differentiable vector field $A$,

The Laplacian of a twice differentiable scalar field is the divergence of its gradient, ${\nabla }^{2}f=\nabla \cdot \nabla f$; it acts componentwise on vector fields.

For a twice differentiable scalar field on the plane,

For twice differentiable scalar fields,

The biharmonic operator applies the Laplacian twice. Under clamped boundary conditions, two integrations by parts give the positive quadratic form

The clamped problem uses the weak solution identity $\int \Delta u\Delta v=\int fv$ for every $v$ in the clamped second-order Sobolev space. For $f\in L^2(U)$, the clamped Hessian identity and the Poincare inequality make this a bounded coercive bilinear form. The Lax-Milgram theorem gives a unique weak solution without a mean-zero condition on $f$.

The biharmonic equation is ${\nabla }^{4}f=0$, where the biharmonic operator is the square of the Laplacian. In planar Stokes flow, taking the curl of the momentum equation and using a Cartesian streamfunction produces this equation. Its decaying spatial Fourier series modes have form $(A+By)e^{-|k|y}{e}^{ikx}$.

For smooth vector fields $F,G$,andBoth follow by contracting two Levi-Civita symbols and applying the product rule.

The displayed vector calculus identity follows by contracting the two alternating tensors in the component formula for curl. If both fields have zero divergence, only the two directional derivatives remain. It rewrites the transport terms in the adjoint linearized Navier-Stokes evolution.

In three dimensions, ${ϵ}_{ijk}$ is zero when indices repeat and is the sign of the permutation $(i,j,k)$ otherwise.

The determinant transformation formula for the Levi-Civita symbol shows that it is invariant under every proper rotation matrix, for which $\det R=1$. Under an orthogonal reflection its sign reverses. This distinguishes proper-rotation isotropy from the pseudotensor transformation convention when both orientations are included.

Contracting one index gives

For three-dimensional vectors,The corresponding identity in ${\mathbb{R}}^n$ is

Contracting two Levi-Civita symbols gives

For linearly independent unit vectors $y,z$ with $c=y\cdot z$, a pairis attained by some unit vector $x$ exactly whenThe left-hand side is the squared norm of the least-norm vector having those two inner products.

A vector field $X$ is conservative when $X=\nabla \Phi$ for a potential of a conservative vector field $\Phi$. On a simply connected open subset of ${\mathbb{R}}^3$, every continuously differentiable curl-free vector field is conservative. Locally this follows from the Poincare lemma applied to the differential one-form $X_{i}d{x}^i$; globally, simple connectivity makes its line integral independent of the path. The fundamental theorem for line integrals expresses that path independence as the difference of the potential at the endpoints.

If a periodic vector field has a periodic potential of a conservative vector field, the line integral between points separated by a period vector must vanish, since it is the potential difference. A periodic gradient field need not satisfy this: the constant field $(1,2)$ on the plane has potential $x+2y$, but its integrals over the two unit translations are $1$ and $2$. On the periodic quotient these translation integrals are nonzero periods, obstructing a single-valued potential there.

A potential for a conservative vector field $X$ is a function with scalar values $\Phi$ whose gradient is $X$. Potentials differ by a constant on each connected component. The fundamental theorem for line integrals gives ${\int }_{\gamma }X\cdot d\mathbf{x}=\Phi (\gamma (1))-\Phi (\gamma (0))$. In mechanics a conservative force is conventionally written $-\nabla V$, so its potential energy is $V=-\Phi$. This spatial potential differs from the scalar potential $V(\varphi )$ that denotes the derivative-free energy density of a scalar field.

A line integral is path independent when its value depends only on the two endpoints. On a connected domain, this holds for the line integral of every exact differential.

For an upward-oriented band of the circular elliptic paraboloid between radii $r_0$ and $R$, the vector field $B=(-y^3,x^3,z^3)$ has vertical curl $3(x^2+y^2)$. Its surface integral is $3\pi (R^4-r_0^4)/2$. Stokes theorem gives the same value from the counterclockwise outer circle minus the counterclockwise inner circle. Equivalently the induced inner boundary direction is clockwise. Forgetting the inner boundary or its reversed orientation changes the answer.

A surface integral uses the area element induced by a parametrisation to integrate over a surface.

The graph $z=g(x,y)$ has upward-oriented vector area element $d\mathbf{S}=(-g_x,-g_y,1)dxdy$ and scalar area element $dS=\sqrt{1+g_x^2+g_y^2}dxdy$. These follow from the cross product of the parametrization derivatives $(1,0,g_x)$ and $(0,1,g_y)$. The vector element is used in flux surface integrals and Stokes theorem; reversing orientation changes its sign.

For an oriented parametrized surface $r(u,v)$, the vector surface element is $d\mathbf{S}=(r_u\times r_v)dudv$, with the sign chosen to match the orientation.

A parametrised surface $r(u,v)$ has area element $|r_u\times r_v|dudv$.

The graph $z=f(x,y)$ has area element $\sqrt{1+f_x^2+f_y^2}dxdy$.

A flux integral ${\int }_{S}F\cdot ndS$ measures flow through an oriented surface.

Green’s theorem converts circulation around a positively oriented planar boundary into the area integral of scalar curl.

A line integral integrates a scalar or tangential vector-field component along a curve.

Applying the divergence theorem componentwise to $T_{ij}{v}_i$ yields a boundary traction term and a volume contraction.

Tensor integration by parts transfers a derivative between tensor factors and introduces the boundary contraction with the outward normal.

The Jacobian matrix contains all first partial derivatives of a coordinate transformation.

To deduce $D{u}_t=I+tDF+o(t)$ from $u_t(x)=x+tF(x)+o(t)$, the remainder must be small in a topology controlling its spatial derivatives, such as $C^1$ locally. Small displacement alone is insufficient: $u_t(x,y,z)=(x+t^2\sin (x/t^2),y,z)$ differs uniformly from the identity by $o(t)$, but has Jacobian determinant $2$ at $x=0$ for nonzero $t$. A smooth flow map generated by a smooth vector field has the required derivative control.

The Jacobian determinant is the determinant of the square Jacobian matrix. Its absolute value is the local volume-scaling factor in the change of variables formula.

For a smooth map $u:{\mathbb{R}}^3\to {\mathbb{R}}^3$, set $V_i={ϵ}_{ijk}{ϵ}_{abc}({\partial }_j{u}_a)({\partial }_k{u}_b)u_c/6$. In Einstein summation convention, applying the product rule leaves only the term differentiating $u_c$: the terms with second derivatives cancel against antisymmetry. The remaining Levi-Civita symbol contraction is $\det Du$, so $\nabla \cdot V=J[u]$. Equivalently this vector field is one third of the cofactor matrix transposed, applied to $u$.

The change-of-variables formula multiplies an integral by the absolute determinant of the Jacobian.

For a smooth local diffeomorphism $a:{\mathbb{R}}^d\to {\mathbb{R}}^d$, each inverse branch contributes an inverse Jacobian determinant in the area formula. Their sum is the weighted inverse multiplicity. A lower determinant bound controls each branch separately, not the number of branches. Bounded weighted inverse multiplicity is sufficient for dispersion with a nonlinear velocity map.

The displayed map is a diffeomorphism of the open unit square onto itself. For fixed $u$, the second coordinate decreases strictly from one to zero as $v$ increases from zero to one. Its inverse is $u=1-x$, $v=(1-y)/[1-(1-x)y]$, and its positive Jacobian determinant is $[1-(1-x)y{]}^2/x$. Composing it with $(t,w)\mapsto ((1-t)/(1-wt),1-w)$ gives determinant $[1-(1-x)y][1-(1-x^2)y{]}^2/[x(1-y)]$. The reciprocal determinant therefore integrates to one over the target square by the change of variables formula, despite boundary singularities.

A hyperbolic change of integration variable uses identities such as $1+{\sinh }^{2}u={\cosh }^{2}u$ to simplify square roots. For an open matter-dominated Friedmann solution, $a=b{\sinh }^2(\eta /2)$ makes the cosmic-time integral elementary.

For a nondecreasing real function $z$ on an interval $I$ and nonnegative measurable function $h$,The pushforward of $z^{\prime }(u)du$ is dominated by Lebesgue measure: on each interval, the integral of the derivative is at most the increase of $z$. This handles jumps and singular parts without a smoothness assumption.

For a locally Lipschitz function $T$ between Euclidean spaces of the same dimension, the area formula integrates its absolute Jacobian determinant against the multiplicity of its fibers. In particular, for an injective $C^1$ map $T:X\to {\mathbb{R}}^d$ and nonnegative measurable $a$,This injective version extends the change of variables formula to maps with a possibly singular derivative.

Polar coordinates use $x=r\cos \theta$, $y=r\sin \theta$ and area element $rdrd\theta$.

The cylindrical coordinate system extends planar polar coordinates by an axial coordinate: $(x,y,z)=(R\cos \varphi ,R\sin \varphi ,z)$.

If the cylindrical components of a vector field have dependence $e^{im\varphi }$, differentiation of the rotating basis adds the displayed cross product term. Scalar-component differentiation alone misses this contribution. For $\mathbf{B}=(J/2)s\hat{\mathbf{\varphi }}$, one also has $(\mathbf{v}\cdot \nabla )\mathbf{B}=(J/2)\hat{\mathbf{z}}\times \mathbf{v}$. These identities are useful in cylindrical magnetohydrodynamic waves.

The cylindrical radius $R=\sqrt{x^2+y^2}$ is the perpendicular distance from the symmetry axis.

A cardioid is an epicycloid with one cusp; a standard polar equation is $r=a(1+\cos \theta )$.

Hyperspherical coordinates extend polar coordinates to $n$ dimensions with one radius and $n-1$ angles.

Spherical coordinates $(r,\theta ,\varphi )$ in ${\mathbb{R}}^3$ use

If orthogonal coordinates $q_i$ satisfythen $h_i=|\partial \mathbf{x}/\partial q_i|$ are their scale factors.

For $\mathbf{A}=A_r{\mathbf{e}}_r+A_{\theta }{\mathbf{e}}_{\theta }+A_{\varphi }{\mathbf{e}}_{\varphi }$, the spherical-coordinate curl follows from the orthogonal-coordinate scale factors $1,r,r\sin \theta$.

The radius-$R$ ball in ${\mathbb{R}}^n$ has volume ${\pi }^{n/2}{R}^n/\Gamma (n/2+1)$.

The boundary of the radius-$R$ ball has area $2{\pi }^{n/2}{R}^{n-1}/\Gamma (n/2)$.

A map is differentiable at $x$ when it differs from an affine map with linear part $D{f}_x$ by $o(\Vert h\Vert )$ at $x+h$.

Functions whose differentials are linearly independent at the point in question. For $F=(F_1,\dots ,F_r)$ this means the derivative has rank $r$, so $F$ is a submersion there. The condition is open. It is distinct from global algebraic independence of functions.

If a differentiable map $F$ takes values in a level set of a differentiable scalar function $H$, the chain rule gives $D{H}_{F(x)}D{F}_x=0$. At a regular point, the image of the derivative lies in the kernel of the nonzero covector $DH$: the tangent space of the level set. For a map into the unit circle this forces rank at most one.

The Frechet derivative of $f$ at $a$ is the unique linear map $D{f}_a$ such that