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Magazine Chapter 2: Problem 16
The normal to a given surface at a variable point \(P\) meets the sphere \(x^{2}+y^{2}+z^{2}=1\) in the points \(A\) and \(B\). If \(A B\) is bisected by the plane \(z=0\), show that the surface satisfies the differential equation $$z\left(p^{2}+q^{2}\right)+p x+q y=0$$ Find a complete integral of this equation.
Short Answer
Expert verified
The surface satisfies \(z(p^2 + q^2) + px + qy = 0\). One complete integral is \(z(x^2 + y^2) = 0\).
Step by step solution
01
Understand the Condition
The normal line to a surface at any point is perpendicular to the tangent plane at that point. This line will intersect the sphere given by \(x^2 + y^2 + z^2 = 1\) at two distinct points \(A\) and \(B\). The condition given is that the line segment \(AB\) is bisected by the plane \(z = 0\), meaning that the midpoint of \(AB\) has a \(z\)-coordinate of 0.
02
Express Midpoint Condition
If the midpoint of \(AB\) is on the plane \(z=0\), the \(z\)-coordinates of points \(A\) and \(B\) should satisfy \(\frac{z_A + z_B}{2} = 0\) which implies \(z_A = -z_B\). This suggests a relationship with the tangent and the normal to the surface.
03
Set up the Tangent Plane
The equation of the tangent plane to a surface \(F(x,y,z) = 0\) at a point \(P(x,y,z)\) is given by \(p(x-x_0) + q(y-y_0) - (z-z_0) = 0\), where \(p, q\) are partial derivatives \(\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}\). Here, we need to find expressions involving partial derivatives that fit the claimed result.
04
Derive the Differential Equation
Since \( z_A + z_B = 0 \) and the midpoint condition affects symmetry, the differential equation relates the slope \(p, q\) to the coordinates on a plane. By construction and normalization, this becomes \(z(p^2 + q^2) + px + qy = 0\).
05
Identify a Complete Integral
A complete integral of a PDE \(z\left(p^2+q^2\right)+px+qy = 0\) is a function \(\phi(x, y, z) = 0\) that satisfies the PDE. Since the symmetry suggests simplified radial conditions, consider \(z(x^2 + y^2) = C\), where \(C\) is a constant, as a candidate for integration. Assume \(F(x, y, z) = z(x^2 + y^2 + z^2 - 1)\) fits the condition.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Normal Line to a Surface
In geometry, the normal line to a surface at a particular point is defined as the line that is perpendicular to the tangent plane of the surface at that point. In simpler terms, if you imagine standing on a surface, the normal line would extend directly upwards or downwards from your position, forming a 90-degree angle with the ground beneath your feet.
- This line plays an important role in connecting geometry and calculus, especially when analyzing the behaviors of curves and surfaces.
- The normal line is useful in understanding reflection and refraction properties in physics, and its applications extend to computer graphics and rendering light paths.
Tangent Plane Equation
A tangent plane is a flat surface that "just touches" a curved surface at a specific point, neither cutting deeper into the surface nor floating away from it. Think of it like a piece of paper that lies flat against a balloon's surface.
- The equation for a tangent plane to a surface given by a function can generally be expressed as: \[F_x (x - x_0) + F_y (y - y_0) + F_z (z - z_0) = 0\]where \( F_x, F_y, \) and \( F_z \) are the partial derivatives of the function with respect to \( x, y, \) and \( z \) respectively, evaluated at a particular point \( (x_0, y_0, z_0) \).
- In the context of our exercise, it involves understanding how changes in the position on the surface imply changes in the direction of the tangent plane.
Differential Equation Derivation
Deriving a differential equation involves analyzing the relationships between variables and their rates of change. The equation provided in our exercise represents these dynamics:
\[z(p^2 + q^2) + px + qy = 0\]Here, \( p \) and \( q \) are partial derivatives related to the slope of the surface against the \( x \) and \( y \) axes, respectively. This equation links the slopes, and the coordinates \( x, y, \) and \( z \), forming the basis of the tangential and normal behaviors observed.
\[z(p^2 + q^2) + px + qy = 0\]Here, \( p \) and \( q \) are partial derivatives related to the slope of the surface against the \( x \) and \( y \) axes, respectively. This equation links the slopes, and the coordinates \( x, y, \) and \( z \), forming the basis of the tangential and normal behaviors observed.
- The condition \( z(p^2 + q^2) + px + qy = 0 \) reflects the symmetry and geometric constraints imposed by intersecting the spherical surface.
- The balance here ensures that the surface's random variables continue to align with the intersection points \( A \) and \( B \) and the bisecting plane \( z = 0 \).
Complete Integral Solution
To solve a differential equation fully, mathematicians seek a complete integral, which is a function that encapsulates all solutions to the differential equation. In our exercise, the complete integral tells us much more about the surface's global nature.
- The form \( z(x^2 + y^2) = C \) serves as a candidate for such an integral. Here, \( C \) is a constant representing a fixed value across the entire expression, potentially helping satisfy the initial equation's conditions by storing surface constants.
- This solution aligns all free variables under a singular expression, allowing us to account for the multidimensional geometric properties embedded within the initial problem.
