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Questions and Answers
What is the main condition for two curves to be considered orthogonal at their intersection?
Which equation properly represents the differential equation relationship for orthogonal trajectories derived from $y = f(x)$?
In the context of orthogonal trajectories, what type of curves can usually be derived from a family of parabolas such as $y = ax^2$?
Which of the following applications is NOT associated with orthogonal trajectories?
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What is the first step in proving that two families of curves are orthogonal?
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What type of differential equations are typically solved to find orthogonal trajectories?
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For the family of circles given by $x^2 + y^2 = r^2$, what type of curves represent their orthogonal trajectories?
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What happens to the slope of the tangent line of an orthogonal trajectory at the point of intersection?
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Study Notes
Definition and Properties
- Orthogonal Trajectories: Curves that intersect a given family of curves at right angles (90 degrees).
- Family of Curves: A set of curves defined by a particular function or equation.
- Intersection Angle: If two curves intersect at a point, the angle between their tangents at that point is 90° for them to be orthogonal.
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Properties:
- If ( y = f(x) ) is a family of curves, the orthogonal trajectories can be found by solving the related differential equations.
- The slopes of the tangent lines of the original curve and its orthogonal trajectory satisfy the condition: ( m_1 \cdot m_2 = -1 ).
Mathematical Proof
- Basic Concept: To prove two families of curves are orthogonal, show that their gradients (derivatives) multiply to -1 at points of intersection.
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Steps:
- Start with the equation of the family of curves ( y = f(x) ).
- Find the derivative ( \frac{dy}{dx} = f'(x) ).
- For orthogonal trajectories, consider the family ( y = g(x) ) and find ( \frac{dy}{dx} = g'(x) ).
- Set up the relationship: ( f'(x) \cdot g'(x) = -1 ).
- Solve the differential equation obtained from this relationship to find ( g(x) ).
Examples and Applications
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Example 1:
- Given the family of circles ( x^2 + y^2 = r^2 ), find orthogonal trajectories.
- Solution involves finding the differential equation of circles, then solving it to derive the family of straight lines.
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Example 2:
- For the family of parabolas ( y = ax^2 ), the orthogonal trajectories can be derived to be linear functions of the form ( y = mx + b ).
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Applications:
- Used in physics to analyze fields (e.g., electric and magnetic fields).
- Important in engineering for flow lines in fluids.
- Applied in biological models to understand population dynamics.
Relation to Differential Equations
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First-Order Differential Equations: Orthogonal trajectories can be found by solving first-order differential equations derived from the original family of curves.
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Methods:
- Convert the given family of curves into a differential equation.
- Use the relationship between the derivatives of the curves (as mentioned in Mathematical Proof).
- Solve the resulting differential equation to obtain the orthogonal trajectories.
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Examples of Differential Equations:
- If the family is defined by ( F(x, y) = 0 ), then the orthogonal trajectories satisfy the equation ( \frac{dy}{dx} = -\frac{F_x}{F_y} ), where ( F_x ) and ( F_y ) are partial derivatives.
Definition and Properties
- Orthogonal trajectories are defined as curves that intersect a family of curves at right angles (90°).
- A family of curves comprises curves governed by a specific function or equation.
- For curves to be orthogonal at an intersection point, the angle between their tangents must be 90°.
- To find orthogonal trajectories of a given family ( y = f(x) ), solve related differential equations.
- The product of the slopes of the tangent lines of an original curve and its orthogonal trajectory must equal -1 (i.e., ( m_1 \cdot m_2 = -1 )).
Mathematical Proof
- To establish that two families of curves are orthogonal, demonstrate that their gradients multiply to -1 at intersection points.
- Start with the equation ( y = f(x) ) for the family of curves.
- Obtain the derivative ( \frac{dy}{dx} = f'(x) ) representing the slope of the original curve.
- For orthogonal trajectories, consider ( y = g(x) ) and find ( \frac{dy}{dx} = g'(x) ).
- Establish the relationship ( f'(x) \cdot g'(x) = -1 ) to connect the two families.
- Solve the derived differential equation to find the equation for the orthogonal trajectories ( g(x) ).
Examples and Applications
- Example 1: For circles described by ( x^2 + y^2 = r^2 ), derive orthogonal trajectories as straight lines by solving the associated differential equation.
- Example 2: From parabolas ( y = ax^2 ), the orthogonal trajectories take the form of linear equations ( y = mx + b ).
- Applications span various fields, such as:
- Physics: Analyzing electric and magnetic field lines.
- Engineering: Understanding fluid flow lines.
- Biology: Modeling population dynamics.
Relation to Differential Equations
- The orthogonal trajectories can be derived using first-order differential equations obtained from the original curves.
- Methods include:
- Translating the family of curves into a differential equation.
- Applying the relationship between derivatives, as established in proof.
- Solving the resulting differential equation to find the orthogonal paths.
- When defined by ( F(x, y) = 0 ), the orthogonal trajectories follow ( \frac{dy}{dx} = -\frac{F_x}{F_y} ) where ( F_x ) and ( F_y ) denote the partial derivatives of the function.
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Description
Explore the concepts of orthogonal trajectories and their properties in calculus. This quiz will test your understanding of how curves intersect and the mathematical proof involved in determining orthogonality. Evaluate your skills in solving related differential equations.
