The metric in surface coordinates

Try

On a cylinder of radius R, let u measure distance around the circumference and v measure height. The parametrization X(u,v)=(Rcos(u/R),Rsin(u/R),v) has orthonormal coordinate derivatives. Its metric is du²+dv², exactly the planar metric, even though the cylinder bends in space.

The first fundamental form is the Euclidean dot product restricted to the tangent plane, expressed in surface coordinates. Its coefficients are E=X_u·X_u, F=X_u·X_v and G=X_v·X_v. The matrix g has rows (E,F) and (F,G).

Predict

Predict whether F=0 means the surface is flat. It only means the two coordinate directions are orthogonal at the point. A sphere has familiar orthogonal coordinates and nonzero Gaussian curvature.

Predict whether the same tangent vector can have different coordinate components in different charts. It can, but the quadratic form must give the same squared length after both the components and metric matrix transform.

Experiment

Inspect two coordinate systems on the same plane.

Coordinates	Metric	Squared speed
Cartesian x,y	dx²+dy²	x′²+y′²
Polar r,θ, with r>0	dr²+r²dθ²	r′²+r²θ′²

A unit angular component in polar coordinates has length r, not one. At r=3, the coordinate vector (0,1) has squared norm nine. Compute the angle between two specified coordinate vectors using aᵀgb divided by the product of their g-norms.

Understand

For a tangent vector V=aX_u+bX_v, expansion gives ||V||²=Ea²+2Fab+Gb². Since X_u,X_v are independent, this is positive for every nonzero pair (a,b). The Gram determinant identity gives EG−F²=||X_u×X_v||²>0. Together with E>0 this characterizes positive definiteness of the two-by-two matrix.

For a curve γ(t)=X(u(t),v(t)), the chain rule yields γ′=X_u u′+X_v v′. Its speed is the square root of E u′²+2F u′v′+G v′², evaluated along the curve. Integrating this speed gives intrinsic curve length.

If old tangent components are a=Jb in new coordinates, then aᵀga=bᵀJᵀgJb. Thus the new metric matrix is JᵀgJ. The numerical squared length is unchanged even though each coefficient may change. This is the basic transformation rule for a covariant bilinear form.

Apply

The cylinder computation provides an important separation. Its metric agrees locally with a plane’s, while its normal varies. The first form records what can be measured along the surface, not all information about how it sits in space. The second form will record bending relative to a normal.

Repair “The coefficient G is the length of a coordinate vector.” It is the squared length; the length is √G. Similarly, F is an unnormalized dot product, not itself an angle.

For optional depth, one can prescribe a smooth positive-definite matrix field as a Riemannian metric without first embedding it in Euclidean space. The later connection and curvature formulas use only this matrix and its derivatives, extending the calculations beyond embedded surfaces.

Continue

When using a coordinate formula, specify whether entries represent vectors, lengths or squared lengths. Continue to surface area and integration to derive the two-dimensional measure associated with the same metric.