If the body is not homogeneous, then the density is a function of the position.

If a mathematical function is used to represent the production function, and if that production function is homogeneous, returns to scale are represented by the degree of homogeneity of the function.

If the Earth were perfectly spherical and homogeneous, then longitude at a point would just be the angle between a vertical north-south plane through that point and the plane of the Greenwich meridian.

If the universe is homogeneous at a large scale, then there would be four times as many stars in a second shell between 2, 000, 000, 000 to 2, 000, 000, 001 light years away.

If two dissimilar homogeneous materials produce thermal EMF1 when the junctions are at T1 and T2 and produce thermal EMF2 when the junctions are at T2 and T3, the EMF generated when the junctions are at T1 and T3 will be EMF1 + EMF2, provided T1 < T2 < T3.

If goods are homogeneous, firms know that a change in their market share is probably due to a price cut ( or quantity increase ) by another member.

If any two functions are solutions to Laplace's equation ( or any linear homogeneous differential equation ), their sum ( or any linear combination ) is also a solution.

If the azeotropic composition is outside the miscibility gap or the constituents of the mixture are completely miscible the type of azeotrope is called a homogeneous azeotrope.

If the supersaturation is high enough, and no suitable surfaces are present, particles may condense in the absence of a pre-existing surface, which is known as homogeneous nucleation.

If the body consists of a homogeneous material with sufficiently known physical properties, the thermal mass is simply the mass of material present times the specific heat capacity of that material.

If I is a homogeneous ideal in A, then is also a graded ring, and has decomposition

If the recurrence is inhomogeneous, a particular solution can be found by the method of undetermined coefficients and the solution is the sum of the solution of the homogeneous and the particular solutions.

If there are multiple consumers with homogeneous demand, then profit will equal n times the area ABC, where n is the number of consumers.

If the sex organs are homogeneous, the individual will not be expected to exhibit any intersex traits.

If the physical formula is not dimensionally homogeneous, and therefore erroneous,

If programmers find that they are using multiple objects in the same way, and often have nearly identical code to handle each of them, then composite is a good choice ; it is less complex in this situation to treat primitives and composites as homogeneous.

If the transformation equations defining the generalized coordinates are independent of t, and the Lagrangian is a sum of products of functions ( in the generalized coordinates ) which are homogeneous of order 0, 1 or 2, then it can be shown that H is equal to the total energy E = T + V.

If all units of land are homogeneous, but demand exceeds supply, the entire land will earn economic rent by virtue of its scarcity.

If V is a projective variety defined by a homogeneous ideal I, then the values for which A or I appear explicitly in previous definitions must be decreased by one.

If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring

If R is a quotient of by a homogeneous ideal, then the canonical surjection induces the closed immersion

If the original equations are given in homogeneous form, these can be obtained by setting z

As before, suppose that E is a fibre bundle with structure group G. In the special case when G has a free and transitive left action on F ′, so that F ′ is a principal homogeneous space for the left action of G on itself, then the associated bundle E ′ is called the principal G-bundle associated to the fibre bundle E. If, moreover, the new fibre F ′ is identified with G ( so that F ′ inherits a right action of G as well as a left action ), then the right action of G on F ′ induces a right action of G on E ′.

If it is homogeneous of degree 1, it exhibits constant returns to scale.

If ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, x < sub > 3 </ sub >) are the Cartesian coordinates and ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, u < sub > 3 </ sub >) are the orthogonal coordinates, then

If certain coordinate systems are used, for instance, polar-toroidal coordinates ( common in plasma physics ) using the notation ∇ × F will yield an incorrect result.

If ( x, y ) are the Cartesian coordinates of a point, then (− x, y ) are the coordinates of its reflection across the second coordinate axis ( the Y axis ), as if that line were a mirror.

To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If ( x, y ) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

If the coordinates represent spatial positions ( displacements ), it is common to represent the vector from the origin to the point of interest as.

If y is a point where the vector field v ( y ) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude.

If instead we use polar coordinates with the origin at one focus, with the angular coordinate still measured from the major axis, the ellipse's equation is

If the coordinates are chosen badly, the laws of motion may be more complex than necessary.

If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion ( known as the Lagrange or Euler – Lagrange equations ) are a set of equations:

If you change units ( a special case of a change of coordinates ) from meters to milimeters, a scale factor of 1 / 1000, a displacement of 1 m becomes 1000 mm – a contravariant change in numerical value.

If a curve is defined in polar coordinates as, then its curvature is

If the manifold M is smooth ( respectively analytic )--- that is, the change of coordinates are smooth ( respectively analytic )--- then one can make sense of the notion of smooth ( respectively analytic ) vector fields.

If the effects of the quantum mechanical nuclear motion are to be studied, for instance because a vibrational spectrum is required, this electronic computation must be in nuclear coordinates.

If Cartesian coordinates in SI units are used, then the components of the position four-vector are given by: x < sup > 0 </ sup > = t, x < sup > 1 </ sup > = x, x < sup > 2 </ sup > = y, and x < sup > 3 </ sup > = z, where t is time in seconds, and x, y, and z are distances in meters.

If a multivariate dataset is visualised as a set of coordinates in a high-dimensional data space ( 1 axis per variable ), PCA can supply the user with a lower-dimensional picture, a " shadow " of this object when viewed from its ( in some sense ) most informative viewpoint.

If this window contains windowless embedded objects, the message may have to be relayed to such an object if the coordinates of the mouse-pointer are over this object.

If those coordinates are placed on a map, they are right outside Secor.

If these texture coordinates are linearly interpolated across the screen, the result is affine texture mapping.

If a point of an object has coordinates ( x, y, z ) then the image of this point ( as reflected from the mirror in y, z plane ) has coordinates (- x, y, z ) – so mirror reflection is a reversal of the coordinate axis perpendicular to the mirror's surface.

If the coordinates are changed, the boundary of the region of space – time over which the Lagrangian is being integrated also changes ; the original boundary and its transformed version are denoted as Ω and Ω ’, respectively.