If is an outward pointing in-plane normal, whereas is the unit vector perpendicular to the plane ( see caption at right ), then the orientation of C is chosen so that a tangent vector to C is positively oriented if and only if forms a positively oriented basis for R < sup > 3 </ sup > ( right-hand rule ).

If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward.

If the orientation of the tangent relative to some starting position is θ ( s ), then ρ ( s ) is defined by the derivative dθ / ds:

If the incident ray is precisely at the critical angle, the refracted ray is tangent to the boundary at the point of incidence.

If M is an open subset of R < sup > n </ sup >, then M is a C < sup >∞</ sup > manifold in a natural manner ( take the charts to be the identity maps ), and the tangent spaces are all naturally identified with R < sup > n </ sup >.

If the tangent space is defined via curves, the map is defined as

If instead the tangent space is defined via derivations, then

If the incircle is tangent to the hypotenuse AB at point P, then denoting the semi-perimeter as s, we have and, and the area is given by

If one considers the flux of the electric field vector, E, for a tube near a point charge in the field the charge but not containing it with sides formed by lines tangent to the field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube.

If f ( x ) is a real-valued function and a and b are numbers with, then the mean value theorem says that under mild hypotheses, the slope between the two points ( a, f ( a )) and ( b, f ( b )) is equal to the slope of the tangent line to f at some point c between a and b. In other words,

If a curve keeps close to the same direction, the unit tangent vector changes very little and the curvature is small ; where the curve undergoes a tight turn, the curvature is large.

If the unit tangent rotates counterclockwise, then k > 0.

If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold.

If P is a common point of two plane algebraic curves X and Y that is a non-singular point of both of them and, moreover, the tangent lines to X and Y at P are distinct then the intersection multiplicity is one.

If the curves X and Y have a common tangent at P then the multiplicity is at least two.

If a surface does not have a tangent plane at a point, it does not have a normal at that point either.

If the normal is constructed as the cross product of tangent vectors ( as described in the text above ), it is a pseudovector.

If two tangent vectors are given

If U is an open contractible subset of M, then there is a diffeomorphism from TU to U × R < sup > n </ sup > which restricts to a linear isomorphism from each tangent space T < sub > x </ sub > U to

If there were a flux across a line, it would necessarily not be tangent to the flow, hence would not be a streamline.

If the line is tangent to the circle, there is one solution, and if the line intersects the circle in two places, there are two solutions.

If the discriminant is equal to 0, then there is a single solution, where the line is tangent to the circle.

It can be used to approximate the tangent to a curve, at some point P. If the secant to a curve is defined by two points, P and Q, with P fixed and Q variable, as Q approaches P along the curve, the direction of the secant approaches that of the tangent at P, ( assuming that the first-derivative of the curve is continuous at point P so that there is only one tangent ).

If he had been `` liquidated '' in some way, he would have become a martyr, a rallying point for people who shared his ideas.

If we examine the three types of change from the point of view of their internal structure we find an additional profound difference between the third and the first two, one that accounts for the notable difference between the responses they evoke.

If he were to go with White, he would be out there two days, not just listening in the dark at some point between here and Papa-san, but moving ever deeper into enemy land -- behind Papa-san -- itself.

If their schedules were to synchronize, there was no point in wasting time.

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

If the argument is accepted as essentially sound up to this point, it remains for us to consider whether the patient's difficulties in orienting himself spatially and in locating objects in space with the sense of touch can be explained by his defective visual condition.

If this attitude is seriously questioned in the Soviet Union, it does not necessarily follow that the majority of the society in which I live is too aware of the necessity for clarity on this ethical as well as aesthetic point of view.

If the Southerners were sufficiently aroused, they could very well cut the Kennedy legislative program to ribbons from their vantage point of committee chairmanships, leaving Sam Rayburn leading a truncated, unworkable party.

If it was, then it must have been God's intention to translate him at a certain point from time to eternity.

If one also removes the second postulate (" a line can be extended indefinitely ") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.

If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their perpendicular height of incidence, i. e. their distance from the axis.

If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence ; the sine condition then becomes sin u ' 1 / h1 = sin u ' 2 / h2.

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 a system consists of several particles, the total angular momentum about a point can be obtained by adding ( or integrating ) all the angular momenta of the constituent particles.

If the net force on some body is directed always toward some fixed point, the center, then there is no torque on the body with respect to the center, and so the angular momentum of the body about the center is constant.

If the score reaches 20-all, then the game continues until one side gains a two point lead ( such as 24 – 22 ), up to a maximum of 30 points ( 30 – 29 is a winning score ).

If the filter shows amplitude ripple within the passband, the x dB point refers to the point where the gain is x dB below the nominal passband gain rather than x dB below the maximum gain.

If the player and dealer have the same point total, this is called a " push " and the player typically doesn't win or lose money on that hand.

If the heat of vaporization and the vapor pressure of a liquid at a certain temperature is known, the normal boiling point can be calculated by using the Clausius-Clapeyron equation thus:

If the large-scale Universe appears isotropic as viewed from Earth, the cosmological principle can be derived from the simpler Copernican principle, which states that there is no preferred ( or special ) observer or vantage point.

If a player is one point away from winning a match, that player's opponent will always want to double as early as possible in order to catch up.

If the holes are closed, the gas will only mix with ambient air at the point of combustion, that is, only after it has exited the tube at the top.