In mathematics, the first fundamental form or metric basic form is a function from the theory of surfaces in three-dimensional Euclidean space , a sub-area of classical differential geometry . The first fundamental form enables, among other things, the following tasks to be dealt with:
Calculation of the length of a curve on the given surface
Calculation of the angle at which two curves intersect on the given surface
Calculation of the area of a patch of the given area
Let a surface be defined by a mapping
on an open subset ${\ displaystyle U \ subset \ mathbb {R} ^ {2}}$
${\ displaystyle X \ colon U \ to \ mathbb {R} ^ {3}, \ quad (u, v) \ mapsto X (u, v)}$
given, i.e. through and
parameterized. For the point of the surface determined by the parameter values and
the coefficients of the first fundamental form are defined as follows:
${\ displaystyle u}$${\ displaystyle v}$${\ displaystyle u}$${\ displaystyle v}$
To simplify matters, one often leaves out the arguments and only writes ,
and for the coefficients. The first fundamental form is then the quadratic form${\ displaystyle E}$${\ displaystyle F}$${\ displaystyle G}$
The numbers are the coefficients of the covariant
metric tensor . So this one has the matrix representation
${\ displaystyle g_ {ij}}$
${\ displaystyle (g_ {ij}) = {\ begin {pmatrix} E&F \\ F&G \ end {pmatrix}}}$.
This tensor , i.e. the bilinear form represented by this matrix , is often referred to as the first fundamental form${\ displaystyle g}$
The following applies to the coefficients of the first fundamental form:
${\ displaystyle E \ geq 0; \ quad G \ geq 0; \ quad EG-F ^ {2} \ geq 0}$.
It is the discriminant (i.e., the determinant of the matrix representation) of the first fundamental form. If it is also valid, it also follows and and the first fundamental form is positive definite . This is the case if and only if and are linearly independent. A surface with a positive definite first fundamental form is called differential geometry regular or differential geometry regular parameterized .
${\ displaystyle EG-F ^ {2}}$${\ displaystyle EG-F ^ {2}> 0}$${\ displaystyle E> 0}$${\ displaystyle G> 0}$${\ displaystyle X_ {u}}$${\ displaystyle X_ {v}}$
Length of a surface curve
A curve on the given surface can be expressed by two real functions
and : The point located on the surface is
assigned to each possible value of the parameter
. If all functions involved are continuously differentiable, the following applies to the length of the
curve segment defined by :
${\ displaystyle \ varphi _ {1}}$${\ displaystyle \ varphi _ {2}}$${\ displaystyle t}$${\ displaystyle X (\ varphi _ {1} (t), \ varphi _ {2} (t))}$${\ displaystyle t \ in [a, b]}$
${\ displaystyle X (u, v) = {\ begin {pmatrix} r \ sin u \ cos v \\ r \ sin u \ sin v \\ r \ cos u \ end {pmatrix}}}$.
For the coefficients of the first fundamental form we get:
${\ displaystyle E = X_ {u} (u, v) \ cdot X_ {u} (u, v) = {\ begin {pmatrix} r \ cos u \ cos v \\ r \ cos u \ sin v \\ -r \ sin u \ end {pmatrix}} \ cdot {\ begin {pmatrix} r \ cos u \ cos v \\ r \ cos u \ sin v \\ - r \ sin u \ end {pmatrix}} = r ^ {2}}$
${\ displaystyle F = X_ {u} (u, v) \ cdot X_ {v} (u, v) = {\ begin {pmatrix} r \ cos u \ cos v \\ r \ cos u \ sin v \\ -r \ sin u \ end {pmatrix}} \ cdot {\ begin {pmatrix} -r \ sin u \ sin v \\ r \ sin u \ cos v \\ 0 \ end {pmatrix}} = 0}$
${\ displaystyle G = X_ {v} (u, v) \ cdot X_ {v} (u, v) = {\ begin {pmatrix} -r \ sin u \ sin v \\ r \ sin u \ cos v \ \ 0 \ end {pmatrix}} \ cdot {\ begin {pmatrix} -r \ sin u \ sin v \\ r \ sin u \ cos v \\ 0 \ end {pmatrix}} = r ^ {2} \ sin ^ {2} u}$
The first fundamental form is therefore
${\ displaystyle ds ^ {2} = r ^ {2} \, du ^ {2} + r ^ {2} \ sin ^ {2} (u) \, dv ^ {2}}$.
Special case graph of a function
If the area under consideration is the graph of a function over the parameter range , i.e. for all , then:
${\ displaystyle f}$${\ displaystyle U}$${\ displaystyle X (u, v) = (u, v, f (u, v))}$${\ displaystyle (u, v) \ in U}$
Here and denote the partial derivatives of to or .
${\ displaystyle f_ {u}}$${\ displaystyle f_ {v}}$${\ displaystyle f}$${\ displaystyle u}$${\ displaystyle v}$