If we use a cylindrical coordinate system with basis vectors, then the gradient of and the divergence of are given by

If there is one particular coordinate change that leads to a decrease in the total energy in both directions, the stationary point is a transition structure and the coordinate is the reaction coordinate.

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 we are to keep a right-handed coordinate system, into the page will be in the negative z direction.

If the axes are named x, y, and z, then the x coordinate is the distance from the plane defined by the y and z axes.

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.

If we view the DFT as just a coordinate transformation which simply specifies the components of a vector in a new coordinate system, then the above is just the statement that the dot product of two vectors is preserved under a unitary DFT transformation.

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 diffusion coefficient is not a constant, but depends upon the coordinate and / or concentration, Fick's Second Law yields

If the transverse axis of any hyperbola is aligned with the x-axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as

If all these hold, then the coordinate systems are said to be in standard configuration.

If the heavier body is much more massive than the smaller, as for a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate and convenient to describe the motion in a coordinate system that is centered on the heavier body, and we say that the lighter body is in orbit around the heavier.

If R = Π < sub > i in I </ sub > R < sub > i </ sub > is a product of rings, then for every i in I we have a surjective ring homomorphism p < sub > i </ sub >: R → R < sub > i </ sub > which projects the product on the i-th coordinate.

If an object moves with angular velocity ω around a circle of radius r centered at the origin of the x-y plane, then its motion along each coordinate is simple harmonic motion with amplitude r and angular frequency ω.

If the origin of the Cartesian coordinate system is taken to be at the point of suspension of the first pendulum, then the centre of mass of this pendulum is at:

If the tower is 50 m high, then the coordinate vector to the top of the tower is r =( 0 ,-50, 50 ).

If and are coordinate vector fields then and therefore the formula simplifies to

She states, " If the Palestinians agree to coordinate with Israel on the evacuation, establish the rule of law, and demonstrate a capacity to govern, the world may be convinced that finally there is a real partner for peace ".

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.

Prior to any measurements being made, the polarizations of the photons are indeterminate ; If a measurement is made on one of the photons using a two-channel polarizer aligned with the axes of the coordinate system, the photon assumes either horizontal or vertical polarization, with 50 % probability for each orientation, and the other two photons immediately assume the identical polarization.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

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

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 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 both input and output are real numbers, this ordered pair can be viewed as the Cartesian coordinates of a point on the graph of the function.

If are the usual orthogonal Cartesian coordinates, the i th row ( i = 1, ..., m ) of this matrix corresponds to the gradient of the i < sup > th </ sup > component function F < sub > i </ sub >:.

If the fluid is incompressible and viscosity is constant across the fluid, the equation governing the shear stress, in the Cartesian coordinate system, is the tensor

If we restrict the local coordinates to be Cartesian coordinates with the same scale centered at the point of interest, the metric tensor is the Kronecker delta.

If a point has the 3D Cartesian coordinates then its transformation for a single direction to a curvilinear reference system of radius R with linear distortion is,

If Nietzsche's response to Cartesian dualism, enlightenment subjectivity ( i. e., Kant ), reductive materialism ( i. e., Marx ), and reductive idealism ( i. e., Hegel ) is not the only nineteenth-century response, it is one of the most effective.

If Cartesian coordinates are used, this equation is modified to incorporate the change in arc length for a spherical gradient, to each physical dimension:

If a standard right-handed Cartesian coordinate system is used, with the x axis to the right and the y axis up, the rotation R ( θ ) is counterclockwise.

If a left-handed Cartesian coordinate system is used, with x directed to the right but y directed down, R ( θ ) is clockwise.

If we take the fixed point as the origin of a Cartesian coordinate system, then every point can be given coordinates as a displacement from the origin.

If we adopt Cartesian coordinates with line element with corresponding wave operator on the flat background, or Minkowski spacetime, so that the line element of the curved spacetime is, then the Ricci scalar of this curved spacetime is just

If you consider an ellipsoid sitting at an angle in a Cartesian grid then you can consider the projection of that ellipse onto the three axes.

If the fluid is incompressible and viscosity is constant across the fluid, the equation governing the shear stress ( in Cartesian coordinates ) is