Such a connection is equivalently specified by a covariant derivative, which is an operator that can differentiate sections of that bundle along tangent directions in the base manifold.

We must then require that the material derivative of the density vanishes, and equivalently ( for non-zero density ) so must the divergence of the fluid velocity:

In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero ( equivalently, the slope is zero ): where the function " stops " increasing or decreasing ( hence the name ).

points x < sub > 0 </ sub > where the derivative in every direction equals zero, or equivalently, the gradient is zero.

Current evidence suggests that the expansion rate of the universe is accelerating, which means that the second derivative of the scale factor is positive, or equivalently that the first derivative is increasing over time.

Due to the commutative property between the derivative operator and the Gaussian smoothing operator, such scale-space derivatives can equivalently be computed by convolving the original image with Gaussian derivative operators.

* Symplectic form ω is preserved by Hamiltonian flow ; or equivalently, Lie derivative

An affine vector is uniquely determined by specifying the values of the vector field and its first covariant derivative ( equivalently, specifying, and ) at any point of.

A homothetic vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative ( equivalently, specifying, and ) at any point of the manifold.

A Killing vector field is uniquely determined by specifying the values of the vector field and its first covariant derivative ( equivalently, specifying and ) at any point ( for every connected component ) of.

or equivalently, integrating by parts, assuming ƒ < sup >( 2p )</ sup > is differentiable again and recalling that the odd Bernoulli numbers are zero:

Note that this derivation does assume that ƒ ( x ) is sufficiently differentiable and well-behaved ; specifically, that ƒ may be approximated by polynomials ; equivalently, that ƒ is a real analytic function of exponential type.

Let ƒ be the fraction of the circumference subtended by the golden angle, or equivalently, the golden angle divided by the angular measurement of the circle.

In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients ( or equivalently — by clearing denominators — with integer coefficients ).

The geometric interpretation of curl as rotation corresponds to identifying bivectors ( 2-vectors ) in 3 dimensions with the special orthogonal Lie algebra so ( 3 ) of infinitesimal rotations ( in coordinates, skew-symmetric 3 × 3 matrices ), while representing rotations by vectors corresponds to identifying 1-vectors ( equivalently, 2-vectors ) and so ( 3 ), these all being 3-dimensional spaces.

Matching the theory's prediction to observational results for planetary orbits ( or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics ), the proportionality constant can be fixed as κ = 8πG / c < sup > 4 </ sup >, with G the gravitational constant and c the speed of light.

or equivalently ( as consistent with the " method " described above ):

Microwaves are radio waves with wavelengths ranging from as long as one meter to as short as one millimetre, or equivalently, with frequencies between 300 MHz ( 0. 3 GHz ) and 300 GHz.

It was later realized that Chari – Nile was a historical artifact of the discovery of the family, and did not reflect an exclusive relationship between these languages, and the group has been abandoned, with its constituents becoming primary branches of Nilo-Saharan — or, equivalently, Chari – Nile and Nilo-Saharan have merged, with the name Nilo-Saharan retained.

Microwaves are electromagnetic waves with wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz ( 0. 3 GHz ) and 300 GHz.

Refraction is described by Snell's law, which states that for a given pair of media and a wave with a single frequency, the ratio of the sines of the angle of incidence θ < sub > 1 </ sub > and angle of refraction θ < sub > 2 </ sub > is equivalent to the ratio of phase velocities ( v < sub > 1 </ sub > / v < sub > 2 </ sub >) in the two media, or equivalently, to the opposite ratio of the indices of refraction ( n < sub > 2 </ sub > / n < sub > 1 </ sub >):

If the string is approximated with 100 discrete mass points one gets the 100 coupled second order differential equations (), () and () or equivalently 200 coupled first order differential equations.

When a codeword is received or read, the device either compares its check value with one freshly calculated from the data block, or equivalently, performs a CRC on the whole codeword and compares the resulting check value with an expected residue constant.

The electric field is a vector field with SI units of newtons per coulomb ( N C < sup >− 1 </ sup >) or, equivalently, volts per metre ( V m < sup >− 1 </ sup >).

Marx sometimes used the term " productive forces " equivalently with " factors of production ;" in Capital, he uses " factors of production ," in his famous Preface to the Critique of Political Economy, he uses " productive forces " ( note that this is in the English versions and may depend on the translation.

( We could have equivalently chosen to measure probabilities from one to infinity, with infinity representing certain falsehood.

Carbonic acid is the organic compound with the formula H < sub > 2 </ sub > CO < sub > 3 </ sub > ( equivalently OC ( OH )< sub > 2 </ sub >).

A Kirkwood gap is a gap or dip in the distribution of main-belt asteroids with semi-major axis ( or equivalently their orbital period ).

Where E is the number of edges in the graph and V is the number of vertices, Kruskal's algorithm can be shown to run in O ( E log E ) time, or equivalently, O ( E log V ) time, all with simple data structures.

An ideal low-pass filter can be realized mathematically ( theoretically ) by multiplying a signal by the rectangular function in the frequency domain or, equivalently, convolution with its impulse response, a sinc function, in the time domain.

Through the observations he made in the following year, Richer determined that the clock was 2½ minutes per day slower than at Paris, or equivalently the length of a pendulum with a swing of one second there was 1¼ Paris lines, or 2. 6 mm, shorter than at Paris.

Formally, two variables are inversely proportional ( or varying inversely, or in inverse variation, or in inverse proportion or in reciprocal proportion ) if one of the variables is directly proportional with the multiplicative inverse ( reciprocal ) of the other, or equivalently if their product is a constant.

An integral quadratic form has integer coefficients, such as x < sup > 2 </ sup > + xy + y < sup > 2 </ sup >; equivalently, given a lattice Λ in a vector space V ( over a field with characteristic 0, such as Q or R ), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning if.

This is a consequence of the tilt of the Earth's rotation with respect to its orbit, or equivalently, the tilt of the ecliptic ( the path of the sun against the celestial sphere ) with respect to the celestial equator.

This means that at very cold temperatures, the change in energy with respect to beta, or equivalently the change in energy with respect to temperature, is also exponentially small.

The logistic map is a polynomial mapping ( equivalently, recurrence relation ) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations.

An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods ; equivalently, it is a function in the function field of an abelian variety.

Koebe proved the general uniformization theorem that if a Riemann surface is homeomorphic to an open subset of the complex sphere ( or equivalently if every Jordan curve separates it ), then it is conformally equivalent to an open subset of the complex sphere.

An antilinear map may be equivalently described in terms of the linear map from to the complex conjugate vector space.

If the vector space is over a field K, then a basis on which the matrix has the required form exists if and only if all eigenvalues of M lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers.

where U ( t ) and I ( t ) are the voltage and current across the Josephson junction, ' is the " phase difference " across the junction ( i. e., the difference in phase factor, or equivalently, argument, between the Ginzburg – Landau complex order parameter of the two superconductors composing the junction ), and I < sub > c </ sub > is a constant, the critical current of the junction.

For example, Poincaré duality of a smooth compact complex manifold X is given by a pairing of singular cohomology with C-coefficients ( equivalently, sheaf cohomology of the constant sheaf C )

In the standard model, the Higgs field is an SU ( 2 ) doublet, a complex spinor with four real components ( or equivalently with two complex components ).

And the act of " conjugation "— that is, passing from a complex number to its conjugate ( or equivalently, < font color =" orange "> reversing the sign of the imaginary part of a complex number </ font >)—< font color =" orange "> is a basic symmetry of the complex number system .</ font > There are other number systems that admit a bewildering collection of symmetries, of internal mirrors.

One can equivalently, through Serre's GAGA, view this as a result in complex algebraic geometry in the Zariski topology.

Its conjugacy classes are labeled by partitions of n. Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions of n. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representations by the same set that parametrizes conjugacy classes, namely by partitions of n or equivalently Young diagrams of size n.

A topological field of sets is a triple where is a topological space and is a field of sets which is closed under the closure operator of or equivalently under the interior operator i. e. the closure and interior of every complex is also a complex.

and showed that the coefficient ring π < sub >*</ sub >( MU ) ( equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds ) is a polynomial ring Zx < sub > 2 </ sub >,... on infinitely many generators x < sub > i </ sub > ∈ π < sub > 2i </ sub >( MU ) of positive even degrees.

where the square root of a complex number is chosen to produce larger absolute value of the denominator, or equivalently, to satisfy:, where denotes real part of a complex number, and is a complex conjugation of ;

Therefore, in principle, one can derive the complex shear modulus ( or equivalently, the complex viscosity ).