If k, m, and n are 1, so that and, then the Jacobian matrices of f and g are.

If the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal.

If analytical expressions are impossible to obtain either the partial derivatives must be calculated by numerical approximation or an estimate must be made of the Jacobian.

If a function is differentiable at a point, its derivative is given in coordinates by the Jacobian, but a function doesn't need to be differentiable for the Jacobian to be defined, since only the partial derivatives are required to exist.

If the Jacobian ( in this context the matrix formed by the complex derivatives ) of a holomorphic function F, defined from an open set U of C < sup > n </ sup > into C < sup > n </ sup >

If a function is differentiable at, then all of the partial derivatives must exist at, in which case the linear map is given by the Jacobian matrix.

If we define the Jacobian matrix

Jacobian conjecture: If J < sub > F </ sub > is a non-zero constant, then F has an inverse function G: k < sup > N </ sup > → k < sup > N </ sup >, and G is regular ( in the sense that its components are given by polynomial expressions ).

If the above errors be eliminated, the two astigmatic surfaces united, and a sharp image obtained with a wide aperture — there remains the necessity to correct the curvature of the image surface, especially when the image is to be received upon a plane surface, e. g. in photography.

If, in an unsharp image, a patch of light corresponds to an object point, the center of gravity of the patch may be regarded as the image point, this being the point where the plane receiving the image, e. g., a focusing screen, intersects the ray passing through the middle of the stop.

If the creditworthiness of the issuer deteriorates ( e. g. rating downgrade ) and its credit spread widens, the bond price tends to move lower, but, in many cases, the call option part of the convertible bond moves higher ( since credit spread correlates with volatility ).

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

If there are g ( E ) dE states with energy E to E + dE, then the Boltzmann distribution predicts a probability distribution for the energy:

If BCG is accidentally given to an immunocompromised patient ( e. g., an infant with SCID ), it can cause disseminated or life-threatening infection.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1 / g ′( f ( 0 )).

If matrix support is polar ( e. g. paper, silica etc.

If the disk was not otherwise prepared with a custom format, ( e. g. for data disks ), 664 blocks would be free after formatting, giving 664 × 254 = 168, 656 bytes ( or almost 165 kB ) for user data.

If G is a group, and g is a fixed element of G, then the conjugation map

If is a constant, the solution is particularly simple, and describes, e. g., if, the exponential decay of radioactive material at the macroscopic level.

If f is a surjection and a ~ b ↔ f ( a ) = f ( b ), then g is a bijection.

Many results about plane figures are proved, e. g., If a triangle has two equal angles, then the sides subtended by the angles are equal.

If the gain is not constant, e. g., by clipping the output signal at the limits of its capabilities, the output signal will be distorted.

If we allow also the system boundary to move ( e. g. due to moving pistons ) we get a rather general form of the first law for open systems.

If there is another functional group at a carbon, it may be named with the Greek letter, e. g., the gamma-amine in gamma-aminobutanoic acid is on the third carbon of the carbon chain attached to the carboxylic acid group.

If f: A < sub > 1 </ sub > → A < sub > 2 </ sub > and g: B < sub > 1 </ sub > → B < sub > 2 </ sub > are morphisms in Ab, then the group homomorphism Hom ( f, g ): Hom ( A < sub > 2 </ sub >, B < sub > 1 </ sub >) → Hom ( A < sub > 1 </ sub >, B < sub > 2 </ sub >) is given by φ g o φ o f. See Hom functor.

If f: X < sub > 1 </ sub > → X < sub > 2 </ sub > and g: Y < sub > 1 </ sub > → Y < sub > 2 </ sub > are morphisms in C, then the group homomorphism Hom ( f, g ): Hom ( X < sub > 2 </ sub >, Y < sub > 1 </ sub >) → Hom ( X < sub > 1 </ sub >, Y < sub > 2 </ sub >) is given by φ g o φ o f.

`` If Blue Throat has his way he'll keep us all cooped up in here for days '', he said.

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

If any one of them has any power to veto the Secretary General's decisions the nature of the organization will have changed.

If this new Bible does not increase in significance by repeated readings throughout the years, it will not survive the ages as has the King James Version.

If the case is thus determined by us to be domestic, the court has no jurisdiction.

If this capacity had not failed them, they would see that their enemy has made a disastrous miscalculation.

If the new Soviet series has followed the general pattern of previous Russian tests, the shots were roughly half fission and half fusion, meaning a fission yield of 30 to 40 megatons thus far.

If the superintendents do not receive more cooperation from Handlers, it has been suggested that licensed Judges also be qualified to judge this Class.

If your state has no provisions for the numbering of pleasure boats, you must apply for a number from the U.S. Coast Guard for any kind of boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.

If design head has a deep cavity, clay lid will be quite thick at this point ; ;

If you run into excess plug fouling on one truck, check to be sure that the rig has a thermostat.

If we manage to keep track of a Bombus queen after she has left her feeding place, we may discover the snug little hideout which she has fixed up for herself when she woke up from her winter sleep.

If cell Af has not previously been assigned as the information cell of a form in the text-form list, it is now assigned as the information cell of Af.

If of the founders of glottochronology Swadesh has escaped our steady plodding, and Lees has repudiated his own share in the founding, that is no reason why we should swerve.

If the transferor has substantial assets other than the claim, it seems reasonable to assume no corporation would be willing to acquire all of its properties in the dim hope of collecting a claim for refund of taxes.

If the patient can perceive figure kinesthetically when he cannot perceive it visually, then, it would seem, the sense of touch has immediate contact with the spatial aspects of things in independence of visual representations, at least in regard to two dimensions, and, as we shall see, even this much spatial awareness on the part of unaided touch is denied by the authors.

If the distant patron of the suburban branch has been frightened away from downtown by traffic problems, however, the city store can only pressure the politicians to do something about the highways or await the completion of the federal highway program.

If a statement has been assigned an address in the index word area

If she has not had such experiences, the female's normal adolescent degree of indecision will be compounded.

If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.

If Barnett doesn't call a special session in 1961, it will be the first year in the last decade that the Legislature has not met in regular or special session.

If the church has followed the plan of cultivation of prospects and carried through a program of membership preparation as outlined earlier in this book, the process of assimilation and growth will be well under way.