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D^*Dpsi=del ^*del psi+1/4Rpsi, where D is the Dirac operator D:Gamma(S^+)->Gamma(S^-), del is the covariant derivative on spinors, and R is the scalar curvature.

Second and higher derivatives of the metric tensor g_(ab) need not be continuous across a surface of discontinuity, but g_(ab) and g_(ab,c) must be continuous across it.

D^*Dpsi=del ^*del psi+1/4Rpsi-1/2F_L^+(psi), where D is the Dirac operator D:Gamma(W^+)->Gamma(W^-), del is the covariant derivative on spinors, R is the scalar curvature, ...

The Lie derivative of tensor T_(ab) with respect to the vector field X is defined by L_XT_(ab)=lim_(deltax->0)(T_(ab)^'(x^')-T_(ab)(x))/(deltax). (1) Explicitly, it is given ...

Also known as the first fundamental form, ds^2=g_(ab)dx^adx^b. In the principal axis frame for three dimensions, ds^2=g_(11)(dx^1)^2+g_(22)(dx^2)^2+g_(33)(dx^3)^2. At ...

The negative pedal curve of a line specified parametrically by x = at (1) y = 0 (2) is given by x_n = 2at-x (3) y_n = ((x-at)^2)/y, (4) which is a parabola.

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.

The links curve is the quartic curve given by the Cartesian equation (x^2+y^2-3x)^2=4x^2(2-x). (1) The area enclosed by the outer envelope is A_(envelope)=1/6(9pi+8) (2) and ...

In space, the only conformal mappings are inversions, similarity transformations, and congruence transformations. Or, restated, every angle-preserving transformation is a ...

For a logarithmic spiral with parametric equations x = e^(bt)cost (1) y = e^(bt)sint, (2) the involute is given by x = (e^(bt)sint)/b (3) y = -(e^(bt)cost)/b, (4) which is ...