Name: perpendicular curves in calculus
Uploaded: 2026-01-29T09:33:09.000Z
Duration: 1 min 26 s
Channel: JoVE

En électromagnétisme, les raies du champ électrique — représentant la direction de la force électrique — sont toujours perpendiculaires aux raies équipotentielles, le long desquelles l’énergie potentielle reste constante.Ces deux ensembles de droites se croisent à angle droit, formant des trajectoires orthogonales.Pour comprendre le concept mathématiquement, considérons une famille de paraboles s’ouvrant le long de l’axe des x, écrite x = ky², où k est une constante.En dérivant l’équation par rapport à x, on obtient la pente de la tangente en un point quelconque sur une parabole.Pour généraliser le résultat à tous les membres de la famille, en éliminant k , on obtient une pente exprimée en termes de x et y.La direction perpendiculaire à cette pente est donnée par le réciproque négatif.En substituant y/2x dans l’expression de la pente et en réarrangeant on obtient l’équation des trajectoires orthogonales des parabolesEn intégrant cette équation, on obtient une expression générale qui montre une famille d’ellipses centrée à l’origine.Ces ellipses intersectent les paraboles à angle droit, reflétant la façon dont les lignes de champ électrique croisent les surfaces équipotentielles dans les systèmes physiques réels.

trajectoires orthogonales

Les trajectoires orthogonales d&#xE9;crivent la relation g&#xE9;om&#xE9;trique entre deux familles de courbes qui se coupent &#xE0; angle droit. Un exemple illustratif concerne une famille de paraboles s&#x2019;ouvrant horizontalement le long de l&#x2019;axe des abscisses. Ces courbes partagent une forme commune mais diff&#xE8;rent par un facteur d&#x2019;&#xE9;chelle, ce qui donne un ensemble de courbes passant toutes par l&#x2019;origine et s&#x2019;&#xE9;largissant selon des rythmes diff&#xE9;rents.
D&#xE9;termination des trajectoires orthogonales
Pour identifier les trajectoires orthogonales de ces paraboles, la premi&#xE8;re &#xE9;tape consiste &#xE0; calculer la pente de la droite tangente &#xE0; chaque courbe. Cette pente exprime &#xE0; quel point la courbe est inclin&#xE9;e en un point donn&#xE9;. Les trajectoires orthogonales doivent avoir des pentes &#xE9;gales &#xE0; l&#x2019;oppos&#xE9; de l&#x2019;inverse de celles des courbes initiales, garantissant ainsi leur intersection perpendiculaire. Cette condition induit une relation diff&#xE9;rentielle entre les variables x et y, qui peut &#xEA;tre s&#xE9;par&#xE9;e et r&#xE9;solue &#xE0; l&#x2019;aide des m&#xE9;thodes classiques du calcul diff&#xE9;rentiel.
Interpr&#xE9;tation g&#xE9;om&#xE9;trique et physique
La r&#xE9;solution de l&#x2019;&#xE9;quation diff&#xE9;rentielle obtenue r&#xE9;v&#xE8;le une nouvelle famille de courbes : des ellipses centr&#xE9;es &#xE0; l&#x2019;origine. Ces ellipses coupent chaque parabole de la famille initiale &#xE0; angle droit, ce qui satisfait la d&#xE9;finition des trajectoires orthogonales. Le sch&#xE9;ma d&#x2019;intersection de ces deux familles illustre un principe plus g&#xE9;n&#xE9;ral que l&#x2019;on retrouve dans diverses disciplines scientifiques.
Applications en physique
Un exemple physique classique se trouve en &#xE9;lectrostatique, o&#xF9; les lignes de champ &#xE9;lectrique coupent les lignes &#xE9;quipotentielles &#xE0; angle droit. Les lignes de champ indiquent la direction de la force exerc&#xE9;e sur une charge, tandis que les lignes &#xE9;quipotentielles repr&#xE9;sentent des points o&#xF9; le potentiel &#xE9;lectrique est constant. Une relation similaire s&#x2019;observe dans les champs gravitationnels, o&#xF9; les lignes repr&#xE9;sentant la direction de la force gravitationnelle coupent perpendiculairement les surfaces d&#x2019;&#xE9;nergie potentielle constante. Ces exemples soulignent l&#x2019;utilit&#xE9; des trajectoires orthogonales pour visualiser et analyser les champs vectoriels et les fonctions de potentiel.

Trajectoires orthogonales

Les trajectoires orthogonales d&#xE9;crivent la relation g&#xE9;om&#xE9;trique entre deux familles de courbes qui se coupent &#xE0; angle droit. Un ...

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Orthogonal Trajectories

perpendicular curves in calculus

Orthogonal trajectories are a pair of curve families that cross at right angles. In this example, the starting family is made of parabolas that open sideways along the x-axis. They share the same shape, but a scaling parameter changes how wide each curve is. Every curve in the family passes through the origin and spreads out at a different rate.

To find the orthogonal trajectories, the slope of each parabola must be calculated first. The slope of a tangent shows how steep the curve is at a point. The new curves must have slopes that are negative reciprocals of the original slopes. That condition makes the two families intersect perpendicularly. It also leads to a differential relationship between x and y that can be separated and solved with standard calculus methods.

Solving that differential equation gives a second family of curves: ellipses centered at the origin. These ellipses cut across the parabolas at right angles, which is the defining feature of orthogonal trajectories. The pattern shows how one family of curves can be matched with another family that meets it perpendicularly.

This idea also appears in physics. In electrostatics, electric field lines cross equipotential lines at right angles. Field lines show the direction of the force on a charge. Equipotential lines mark places with the same electrical potential. A similar pattern appears in gravitational fields, where lines of gravitational force intersect surfaces of equal potential energy perpendicularly. These examples show how orthogonal trajectories help people visualize and study vector fields and potential functions.

Perpendicular Curves in Calculus

introduction to differential equations

Modeling Motion with Second-Order Equations

Differential equations connect a function with its derivatives. They are used to describe changing systems in science and engineering. A differential equation is classified by order, which is the highest derivative it contains. A first-order equation includes only the first derivative, while a second-order equation includes the second derivative of the unknown function. A common second-order differential equation comes from a spring-mass system. In this idealized model, a mass is attached to a...

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Population growth can be modeled with differential equations by treating population size, P(t), as a function of time. The derivative of P(t) shows how fast the population changes. A simple starting idea is that the growth rate is proportional to the population size. That assumption gives an exponential growth model. In this model, the population rises continuously and faster over time. It can be a useful first approximation, but it does not match long-term behavior in nature. Real...

separable differential equations

Solving Separable Differential Equations

A separable differential equation is a first-order differential equation that can be written with dy/dx as a product of one function of x and one function of y. This form makes it possible to separate the variables so that all y terms stay on one side and all x terms stay on the other. After the variables are separated, each side can be integrated with respect to its own variable. The result is a relationship between x and y. Depending on the functions involved, that relationship may be...

orthogonal trajectories

Orthogonal trajectories are a pair of curve families that cross at right angles. In this example, the starting family is made of parabolas that open sideways along the x-axis. They share the same shape, but a scaling parameter changes how wide each curve is. Every curve in the family passes through the origin and spreads out at a different rate. To find the orthogonal trajectories, the slope of each parabola must be calculated first. The slope of a tangent shows how steep the curve is at a...

linear differential equations

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The integrating factor method solves first-order linear differential equations that are not separable. It is especially useful when a system has a constant driving force and a resistive force that depends on the changing variable. In these problems, the equation can be rearranged so the left-hand side becomes a product rule after multiplication by an integrating factor. A clear example is a car moving under a constant engine force while air resistance acts against its motion. Newton’s second...

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Differential Equations and Falling Motion

Differential equations can model the motion of a falling object when gravity and air resistance act at the same time. In this example, a heavy test weight is released from rest during a ship safety check. Gravity pulls the weight downward, and air resistance pushes upward as the speed increases. Newton’s Second Law links the net force on the weight to its acceleration. Gravity provides a constant downward force equal to mass times gravitational acceleration. Air resistance is modeled as a...

key terms and concepts

Learn how orthogonal trajectories work by finding a family of ellipses that meets sideways parabolas at right angles, with physics examples.

Perpendicular Curves in Calculus | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

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Perpendicular Curves in Calculus

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