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

In electromagnetism, electric field lines&#8212;representing the direction of the electric force&#8212;are always perpendicular to equipotential lines, along which potential energy remains constant.These two sets of lines intersect at right angles, forming orthogonal trajectories.To understand the concept mathematically, consider a family of parabolas opening along the x-axis, written as x = ky&#178;, where k is a constant.Differentiating the equation with respect to x gives the slope of the tangent at any point on a parabola.To generalize the result for all members of the family, eliminating k gives a slope expressed in terms of x and y.The perpendicular direction to this slope is given by the negative reciprocal.Substituting y/2x in the slope expression and rearranging gives the equation for the orthogonal trajectories of the parabolasIntegrating this equation gives a general expression that shows a family of ellipses centered at the origin.These ellipses intersect the parabolas at right angles, mirroring the way electric field lines intersect equipotential surfaces in real physical systems.

orthogonal trajectories

Orthogonal trajectories describe the geometric relationship between two families of curves that intersect each other at right angles. One illustrative case involves a family of parabolas that open sideways along the x-axis. These curves share a common shape but differ by a scaling parameter, resulting in a set of curves that all pass through the origin and widen at different rates.Determining Orthogonal TrajectoriesTo identify the orthogonal trajectories for these parabolas, the first step involves calculating the slope of the tangent to each curve. This slope expresses how steeply the curve rises or falls at a given point. Orthogonal trajectories must have slopes that are the negative reciprocals of the original curves' slopes, ensuring the curves intersect perpendicularly. This condition leads to a differential relationship between the x and y variables that can be separated and solved using standard methods in calculus.Geometric and Physical InterpretationSolving the resulting differential equation reveals a new family of curves: ellipses centered at the origin. These ellipses intersect each parabola in the original family at right angles, fulfilling the definition of orthogonal trajectories. The intersection pattern of these two families illustrates a broader principle seen across various scientific disciplines.Applications in PhysicsA classic physical example can be seen in electrostatics, where electric field lines intersect equipotential lines at right angles. The field lines show the direction of the force exerted on a charge, while equipotential lines represent locations of constant electrical potential. A similar relationship is seen in gravitational fields, where lines representing the direction of gravitational force intersect surfaces of equal potential energy perpendicularly. These examples underscore the utility of orthogonal trajectories in visualizing and analyzing vector fields and potential functions.

Orthogonal Trajectories

Orthogonal trajectories describe the geometric relationship between two families of curves that intersect each other at right angles. One illustrative ...

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Calculus

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Differential Equations

KEY TERMS AND CONCEPTS

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

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modeling with differential equations

Population Growth: Exponential vs. Logistic

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 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

Integrating Factors for Motion with Drag

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...

differential equations: problem solving

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.

calculus

Perpendicular Curves in Calculus

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