Question

Write an equation for the plane tangent to the surface \(F(x, y, z)=0\) at the point \((a, b, c)\)

Short answer

Answer: To find the equation of the tangent plane, follow these steps: 1. Find the gradient vector (normal vector) of the function at the point (a, b, c) by calculating the partial derivatives of F with respect to x, y, and z, and evaluating them at the point (a, b, c). 2. Use this gradient vector as the normal vector to the tangent plane, since they are orthogonal. 3. Write the equation of the tangent plane using the point-normal form with the normal vector and the given point. The equation will be: \(\frac{\partial F}{\partial x}(a, b, c)(x - a) + \frac{\partial F}{\partial y}(a, b, c)(y - b) + \frac{\partial F}{\partial z}(a, b, c)(z - c) = 0\)

Step-by-step solution

Find the gradient vector (normal vector) of the function at the point (a, b, c).

To find the gradient vector of \(F(x, y, z)\) at point \((a, b, c)\), we need to find the partial derivatives of \(F\) with respect to \(x, y, z\) and evaluate them at \((a, b, c)\). The gradient vector is given by \(\nabla F = \left(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z}\right)\). Afterwards, plug in the point \((a, b, c)\) into the gradient vector.

Find the normal vector to the tangent plane.

Since the gradient vector is orthogonal to the level surfaces of the function, it will also be orthogonal (i.e., normal) to the tangent plane. So, our normal vector at the point \((a, b, c)\) is given by \(\textbf{n} = \nabla F(a, b, c)\).

Write the equation of the tangent plane.

We will use the point-normal form of the equation of a plane to find the desired equation. With the normal vector \(\textbf{n} = \left(\frac{\partial F}{\partial x}(a, b, c), \frac{\partial F}{\partial y}(a, b, c), \frac{\partial F}{\partial z}(a, b, c)\right)\) and the point \((a, b, c)\), the equation of the tangent plane can be written as: \(\frac{\partial F}{\partial x}(a, b, c)(x - a) + \frac{\partial F}{\partial y}(a, b, c)(y - b) + \frac{\partial F}{\partial z}(a, b, c)(z - c) = 0\) Now you have the equation of the tangent plane to the surface \(F(x, y, z) = 0\) at the point \((a, b, c)\).

Key concepts

Gradient Vector

When dealing with surfaces and their tangent planes, the gradient vector is a crucial concept. It offers a direction of maximum increase for a multi-variable function like \( F(x, y, z) \). To obtain this vector, you need to compute the partial derivatives of the function with respect to each of its variables \( x, y, \) and \( z \). Mathematically, this is expressed as \( abla F = \left( \frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z} \right) \). The symbol \( abla F \) denotes the gradient vector.
The important property of the gradient vector in the context of tangent planes is that it is perpendicular to the level surfaces of a function. This unique quality is what makes the gradient vector an ideal normal vector for the tangent plane. When evaluating the gradient at a specific point, say \( (a, b, c) \), you simply plug these coordinates into your partial derivatives to get your valued gradient vector \( abla F(a, b, c) \).

Normal Vector

The normal vector is fundamental in establishing a plane, specifically the tangent plane to a surface. A normal vector is the vector that is perpendicular to a surface at a given point. In the context of our equation \( F(x, y, z) = 0 \), the normal vector to the tangent plane at any point \((a, b, c)\) can be directly derived from the gradient vector.
In essence, once the gradient vector \( abla F \) is evaluated at the point \((a, b, c)\), it directly serves as the normal vector \( \textbf{n} = abla F(a, b, c) \). This is because the gradient vector is always orthogonal to the level surfaces of the function, making it an automatic candidate for the role of the normal vector. This orthogonality is the key reason it is used in defining the tangent plane.

• A normal vector gives us the necessary orientation needed to establish the tangent plane.
• It ensures that the plane is tangent and not skewed at any other angle to the point \((a, b, c)\).

Partial Derivatives

Partial derivatives are one of the fundamental building blocks for understanding changes in multivariable functions. When you compute the partial derivative of \( F(x, y, z) \) with respect to \( x \), \( y \), or \( z \), you are finding out how the function changes as one of these variables changes, while the others are kept constant.
For instance, the partial derivative \( \frac{\partial F}{\partial x} \) tells you about the rate of change of the function as only \( x \) is varied. The process is analogous for \( y \) and \( z \). These partial derivatives form the components of the gradient vector, \( abla F \), which is crucial for defining both the tangent plane and the normal vector.

• Partial derivatives help in constructing the gradient vector.
• They offer an insight into the behavior of a function in a localized manner.
• These derivatives serve as the coefficients in the equation of the tangent plane.

Understanding these derivatives helps in analyzing the local behavior of the surface, which in turn, aids in the construction of tangent planes.