حساب دیفرانسیل و انتگرال چندمتغیره

Evaluating an Iterated Double Integral

Compute an iterated double integral from the inside out, treating the outer variable as a constant during the inner integration.

double-integraliterated-integralfubiniintegration-by-parts

Minimum Distance from a Quadric Surface to a Plane

Minimize the plane-distance formula over the surface, often collapsing to an algebraic inequality.

minimum-distancelagrange-multipliersquadric-surfaceplane

Radius of Curvature of a Parametric Plane Curve

For a plane curve r(t)=(x(t),y(t)) , the radius of curvature is R=1/κ with κ= x'y''-y'x'' /((x')^2+(y')^2)^(3/2) .

curvatureradius-of-curvatureosculating-circleparametric-curve

قضیهریاضی عمومی

Stokes' Theorem for Line Integrals

Convert a hard closed line integral into a curl flux through any bounding surface.

stokes-theoremline-integralcurlsurface-integral

Arc Length and the Catenary

The arc length of y=f(x) is ∫√(1+(y')^2) dx ; for the catenary y=acosh(x/a) the identity 1+sinh^2=cosh^2 erases the square root, giving length asinh(x_1/a) .

arc-lengthcatenaryhyperbolic-functionscosh

Equation of a Plane Through a Line and an External Point

A line plus a point not on it fixes a plane: build two in-plane directions, cross them for the normal, and write the point-normal equation.

planecross-productnormal-vectorline-in-space

Volume by Cross-Sections: The Napkin-Ring Problem

Slice a solid perpendicular to one axis, integrate the cross-sectional area A(y); for a sphere drilled through its center, the leftover ring's volume depends only on the ring's height, not the sphere's radius.

volumecross-sectionssolid-of-revolutionwasher-method

قضیهریاضی عمومی

Divergence (Gauss) Theorem for Flux Through a Closed Surface

The outward flux of a vector field through a closed surface equals the triple integral of its divergence over the enclosed solid.

divergence-theoremgauss-theoremfluxsurface-integral

Gradient: Direction and Rate of Steepest Ascent

At every point the gradient points toward the fastest increase, and its magnitude is exactly that maximum rate of change.

gradientdirectional-derivativesteepest-ascentmultivariable-calculus

Implicit Function Theorem: Partial Derivatives from a System of Equations

When a system implicitly defines several variables in terms of others, differentiate the equations in place and solve a small linear system (Jacobian) for the partials — no explicit formulas needed.

implicit-function-theoremjacobiansystem-of-equationspartial-derivatives

Change of Variables in a PDE via the Multivariable Chain Rule

How to transform a second-order PDE under a linear change of variables: build the first-order operators from the chain rule, apply them twice to get the second derivatives, and substitute to simplify (e.g. the wave equation into canonical form).

pdechain-rulechange-of-variableswave-equation

Triple Integrals in Cylindrical Coordinates: Moment of Inertia

For a solid with rotational symmetry, switch to cylindrical coordinates ( dV=r dz dr dθ ) and integrate ρ r^2 to get the moment of inertia I_z about the axis.

moment-of-inertiatriple-integralcylindrical-coordinatessolid-of-revolution

Surface Area of a Solid of Revolution

Revolving a curve sweeps a surface whose area is ∫ 2π(radius) ds , where ds is arc length — pick the variable so the radius is the distance to the axis.

surface-of-revolutionsurface-areaarc-lengthsolid-of-revolution

Mass of a Solid as a Triple Integral of the Density

The mass of a solid is M=iiint_V ρ dV — integrate the (possibly variable) density over the region; when ρ is constant this collapses to ρcdotvolume .

massdensitytriple-integralcentroid

Nonexistence of a Two-Variable Limit by the Path Test

A limit of f(x,y) as (x,y)→(a,b) exists only if it equals the same value along every path to (a,b) . Finding two paths that give different values proves the limit does not exist.

multivariable-limitpath-testnonexistencecalculus