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The slope field can be defined for the following type of differential equations ′ = (,), 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. [3]
When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence. The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are linearly dependent. For example ...
The use of multiple representations supports and requires tasks that involve decision-making and other problem-solving skills. [2] [3] [4] The choice of which representation to use, the task of making representations given other representations, and the understanding of how changes in one representation affect others are examples of such mathematically sophisticated activities.
1.3 Curves with genus greater than one. 1.4 Curve families with variable genus. 2 Transcendental curves. Toggle Transcendental curves subsection. 2.1 Spirals.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Using Newton's identities, it is straightforward to express them in terms of the elementary symmetric functions of the roots, giving =, = +, with e 1 = 0, e 2 = p and e 3 = −q in the case of a depressed cubic, and e 1 = − b / a , e 2 = c / a and e 3 = − d / a , in the general case.