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The slope field can be defined for the following type of differential equations y ′ = f ( x , y ) , {\displaystyle y'=f(x,y),} which can be interpreted geometrically as giving the slope of the tangent to the graph of the differential equation's solution ( integral curve ) at each point ( x , y ) as a function of the point coordinates.
Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.
Placing position on the y-axis and time on the x-axis, the slope of the curve is given by: = =. Here is the position of the object, and is the time. Therefore, the slope of the curve gives the change in position divided by the change in time, which is the definition of the average velocity for that interval of time on the graph.
1.1.1.7 Curve families of variable ... and applied mathematics), physics, engineering, economics, medicine ... online at Google Books This page was last edited ...
Generally, an isocline will itself have the shape of a curve or the union of a small number of curves. Isoclines are often used as a graphical method of solving ordinary differential equations. In an equation of the form y' = f(x, y), the isoclines are lines in the (x, y) plane obtained by setting f(x, y) equal to a constant. This gives a ...
l = slope length α = angle of inclination. The grade (US) or gradient (UK) (also called stepth, slope, incline, mainfall, pitch or rise) of a physical feature, landform or constructed line is either the elevation angle of that surface to the horizontal or its tangent. It is a special case of the slope, where zero indicates horizontality. A ...
A space curve is a curve for which is at least three-dimensional; a skew curve is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to real algebraic curves , although the above definition of a curve does not apply (a real algebraic curve may be disconnected ).
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.