Since at this stress the slope of the material's stress-strain curve, E < sub > t </ sub > ( called the tangent modulus ), is smaller than that below the proportional limit, the critical load at inelastic buckling is reduced.

In solid mechanics, the slope of the stress-strain curve at any point is called the tangent modulus.

Incorporation of reinforcing fillers into the polymer blends also increases the storage modulus at an expense of limiting the loss tangent peak height.

Let γ be a differentiable curve in M with initial point γ ( 0 ) and initial tangent vector X

The path is closed since initial and final directions of the light coincide, and the polarization is a vector tangent to the sphere.

In the Edgeworth box, it is a point at which Octavio ’ s indifference curve is tangent to Abby ’ s indifference curve, and it is inside the lens formed by their initial allocations.

The initial tangent vector is parallel transported to each tangent along the curve ; thus the curve is, indeed, a geodesic.

In either case, df < sub > x </ sub > is a linear map on T < sub > x </ sub > M and hence it is a tangent covector at x.

Just as every differentiable map f: M → N between manifolds induces a linear map ( called the pushforward or derivative ) between the tangent spaces

which has the intuitive interpretation ( see Figure 1 ) that the tangent line to f at a gives the best linear approximation

Every smooth ( or differentiable ) map φ: M → N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:

The technical statement is as follows: if M is a given m-dimensional Riemannian manifold ( analytic or of class C < sup > k </ sup >, 3 ≤ k ≤ ∞), then there exists a number n ( with n ≤ m ( 3m + 11 )/ 2 if M is a compact manifold, or n ≤ m ( m + 1 )( 3m + 11 )/ 2 if M is a non-compact manifold ) and an injective map ƒ: M → R < sup > n </ sup > ( also analytic or of class C < sup > k </ sup >) such that for every point p of M, the derivative dƒ < sub > p </ sub > is a linear map from the tangent space T < sub > p </ sub > M to R < sup > n </ sup > which is compatible with the given inner product on T < sub > p </ sub > M and the standard dot product of R < sup > n </ sup > in the following sense:

To build a bijection from T to R: start with the tangent function tan ( x ), which provides a bijection from (− π / 2, π / 2 ) to R. Next observe that the linear function h ( x ) = πx-π / 2 provides a bijection from ( 0, 1 ) to (− π / 2, π / 2 ).

A related notion of curvature is the shape operator, which is a linear operator from the tangent plane to itself.

For each pair of tangent vectors u, v, R ( u, v ) is a linear transformation of the tangent space of the manifold.

Therefore all tangent vectors in a point p span a linear space, called the tangent space at point p. For example, taking a 2-dimensional space, like the ( curved ) surface of the Earth, its tangent space at a specific point would be the flat approximation of the curved space.

Any linear functional α on tangent vectors can be expanded in terms of the dual basis θ

A tangent vector is by definition a vector that is a linear combination of the coordinate partials.

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

Schuh observed that in a boundary-layer, u is again a linear function of y, but that in this case, the wall tangent is a function of x.

Let denote the tangent space of M at a point p. For any pair of tangent vectors at p, the Ricci tensor evaluated at is defined to be the trace of the linear map given by

Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to R whose restriction to each fibre is a linear functional on the tangent space.

Connections in this sense generalize, to arbitrary vector bundles, the concept of a linear connection on the tangent bundle of a smooth manifold, and are sometimes known as linear connections.

where X and Y are tangent vector fields on M and s is a section of E. One must check that F < sup >∇</ sup > is C < sup >∞</ sup >- linear in both X and Y and that it does in fact define a bundle endomorphism of E.

# for all p in P, the restriction of η defines a linear isomorphism from the tangent space T < sub > p </ sub > P to.

A large portion of U. S. Highway 460 in eastern Virginia ( between Petersburg and Suffolk ) parallels the 52-mile tangent railroad tracks that Mahone had engineered, passing through some of the towns he and Otelia are believed to have named.

A large portion of U. S. Highway 460 between Petersburg and Suffolk, parallels the 52 mile tangent line of the railroad William engineered.

It featured an innovative and durable roadbed through a portion of the Great Dismal Swamp and an arrow-straight 52-mile tangent between Suffolk and Petersburg.

His application of reference lines, a diameter and a tangent is essentially no different than our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates.

In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity.

* The start ( end ) of the curve is tangent to the first ( last ) section of the Bézier polygon.

The tension at c is tangent to the curve at c and is therefore horizontal, and it pulls the section to the left so it may be written (− T < sub > 0 </ sub >, 0 ) where T < sub > 0 </ sub > is the magnitude of the force.

The radius of curvature of the path is ρ as found from the rate of rotation of the tangent to the curve with respect to arc length, and is the radius of the osculating circle at position s. The unit circle on the left shows the rotation of the unit vectors with s.

The required distance ρ ( s ) at arc length s is defined in terms of the rate of rotation of the tangent to the curve, which in turn is determined by the path itself.

This displacement is necessarily tangent to the curve at s, showing that the unit vector tangent to the curve is:

Using the tangent vector, the angle θ of the tangent to the curve is given by:

At each point, the derivative of is the slope of a Line ( geometry ) | line that is tangent to the curve.

The line is always tangent to the blue curve ; its slope is the derivative.

For a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance.

( which did not exist in Diophantus's time ), his method would be visualised as drawing a tangent to a curve at a known rational point, and then finding the other point of intersection of the tangent with the curve ; that other point is a new rational point.

* Spherical helix, tangent indicatrix of a curve of constant precession

As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point.

At each point, the derivative is the slope of a Line ( geometry ) | line that is tangent to the curve.

The line is always tangent to the blue curve ; its slope is the derivative.