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The indices of a contravariant tensor A^j can be lowered, turning it into a covariant tensor A_i, by multiplication by a so-called metric tensor g_(ij), e.g., g_(ij)A^j=A_i.

The illusion shown above which was discovered by Helmholtz in the 19th century. Despite the fact that the two above figures are identical in size, the white hole looks bigger ...

The repeated application of a transformation.

A geodesic triangle with oriented boundary yields a curve which is piecewise differentiable. Furthermore, the tangent vector varies continuously at all but the three corner ...

The Koch snowflake is a fractal curve, also known as the Koch island, which was first described by Helge von Koch in 1904. It is built by starting with an equilateral ...

The second-order ordinary differential equation y^('')+k/xy^'+epsilony^'y=0.

A three-dimensional coordinate system in which the axes do not satisfy the right-hand rule.

Let suma_k and sumb_k be two series with positive terms and suppose lim_(k->infty)(a_k)/(b_k)=rho. If rho is finite and rho>0, then the two series both converge or diverge.

For two lines in the plane with endpoints (x_1,x_2) and (x_3,x_4), the angle between them is given by costheta=((x_2-x_1)·(x_4-x_3))/(|x_2-x_1||x_4-x_3|). (1) The angle theta ...

Two curves phi and psi satisfying phi+psi=0 are said to be linearly dependent. Similarly, n curves phi_i, i=1, ..., n are said to be linearly dependent if sum_(i=1)^nphi_i=0.