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A horizontal line is a straight, flat line that goes from left to right. Given a function (i.e. from the real numbers to the real numbers), we can decide if it is injective by looking at horizontal lines that intersect the function's graph. If any horizontal line intersects the graph in more than one point, the function is not injective.
The silk on a spider's web forming multiple elastic catenaries. In physics and geometry, a catenary (US: / ˈkætənɛri / KAT-ən-err-ee, UK: / kəˈtiːnəri / kə-TEE-nər-ee) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
Bump function. In mathematics, a bump function (also called a test function) is a function on a Euclidean space which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain forms a vector space, denoted or The dual space of this space endowed with a ...
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.
The Stokes stream function, named after George Gabriel Stokes, [ 2 ] is defined for incompressible, three-dimensional flows with axisymmetry. The properties of stream functions make them useful for analyzing and graphically illustrating flows. The remainder of this article describes the two-dimensional stream function.
In physics, the line of action (also called line of application) of a force (F→) is a geometric representation of how the force is applied. It is the straight line through the point at which the force is applied, and is in the same direction as the vector F→. [1][2] The concept is essential, for instance, for understanding the net effect of ...
Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by , defined as the curl of the velocity field describing the continuum motion. In Cartesian coordinates: In words, the vorticity tells how the velocity vector changes when one moves by an infinitesimal distance in a direction perpendicular to it.
The formula is. where is the applied tension on the line, is the resulting force exerted at the other side of the capstan, is the coefficient of friction between the rope and capstan materials, and is the total angle swept by all turns of the rope, measured in radians (i.e., with one full turn the angle ). For dynamic applications such as belt ...