Question

Suppose \(\mathbf{n}\) is a vector normal to the tangent plane of the surface \(F(x, y, z)=0\) at a point. How is \(\mathbf{n}\) related to the gradient of \(F\) at that point?

Short answer

Answer: The normal vector to the tangent plane of a surface \(F(x, y, z) = 0\) at a point is equal to the gradient of \(F\) at that point.

Step-by-step solution

Find the gradient of F

To find the gradient of \(F\), we take the partial derivatives of \(F\) with respect to each of its variables, \(x\), \(y\), and \(z\). The gradient, denoted as \(\nabla F\), is given by: $$\nabla F = \left(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z}\right)$$

Find the tangent plane of F

To find the tangent plane of \(F\) at a point \((x_0, y_0, z_0)\) on the surface, we need to find the linear approximation of \(F\) in the neighborhood of the point. Using the gradient, we can write the equation for the tangent plane as: $$F(x_0, y_0, z_0) + \left(\frac{\partial F}{\partial x}\right)(x-x_0) + \left(\frac{\partial F}{\partial y}\right)(y-y_0) + \left(\frac{\partial F}{\partial z}\right)(z-z_0) = 0$$

Find the normal vector of the tangent plane

The normal vector \(\mathbf{n}\) of the tangent plane can be expressed as the coefficients of the linear terms in the tangent plane equation. Recall that \(\nabla F\) contains these coefficients. Thus, we can conclude that the normal vector \(\mathbf{n}\) is equal to the gradient of \(F\) at the point \((x_0, y_0, z_0)\): $$\mathbf{n} = \nabla F$$ In conclusion, the normal vector of a tangent plane to the surface \(F(x, y, z) = 0\) at a point is related to the gradient of \(F\) at that point by being equal to each other.

Key concepts

Gradient of a Function

Understanding the gradient of a function is essential for visualizing the geometry of surfaces defined by mathematical equations. The gradient represents the slope or steepness of a function at a particular point and is denoted by the symbol abla (nabla).

In the context of a three-dimensional space and a function F(x, y, z), the gradient is a vector that points in the direction of the steepest ascent on the surface described by the equation F(x, y, z) = 0. To find this vector, we compute the partial derivatives of F with respect to each variable—x, y, and z.

These derivatives express how the function changes as we move along the respective axes:

Normal Vector

A normal vector, often represented by \(\mathbf{n}\), plays a pivotal role in understanding geometric surfaces. It is a vector that is perpendicular to the surface at a specific point. For a tangent plane, which is a flat approximation of a surface at a given point, the normal vector defines its orientation in three-dimensional space.

When dealing with the surface F(x, y, z) = 0, the normal vector at any point on this surface can be found using the gradient of F. This is because the gradient is always perpendicular to the level curves of F, which means that it is also perpendicular to the tangent plane formed at the point of contact. Thus, the normal vector \(\mathbf{n}\) essentially has the same components as the gradient vector of F at the given point.

Partial Derivatives

Partial derivatives represent how a function changes as we vary one of its variables while keeping the others constant. They are fundamental in multivariable calculus and in defining the gradient, as well as the tangent plane and normal vector.

For a function F(x, y, z), we can take partial derivatives with respect to x, y, and z. These partial derivatives indicate the rate of change of F as we move in the direction of each axis separately. When we combine these derivatives into a vector, we get the gradient, which not only shows the direction of the steepest ascent but also provides crucial information about the tangent plane and normal vector at any point on the surface F(x, y, z) = 0.