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Chapter 15: Problem 1

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

Expert verified
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

01

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: $$\nabla F = \left(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z}\right)$$
02

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: $$F(x_0, y_0, z_0) + \left(\frac{\partial F}{\partial x}\right)(x-x_0) + \left(\frac{\partial F}{\partial y}\right)(y-y_0) + \left(\frac{\partial F}{\partial z}\right)(z-z_0) = 0$$
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥�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.

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Most popular questions from this chapter

The level curves of the surface \(z=x^{2}+y^{2}\) are circles in the \(x y\) -plane centered at the origin. Without computing the gradient, what is the direction of the gradient at (1,1) and (-1,-1) (determined up to a scalar multiple)?

Heron's formula The area of a triangle with sides of length \(a, b\) and \(c\) is given by a formula from antiquity called Heron's formula: $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ where \(s=\frac{1}{2}(a+b+c)\) is the semiperimeter of the triangle. a. Find the partial derivatives \(A_{\sigma}, A_{b},\) and \(A_{c}\) b. A triangle has sides of length \(a=2, b=4, c=5 .\) Estimate the change in the area when \(a\) increases by \(0.03, b\) decreases by \(0.08,\) and \(c\) increases by 0.6 c. For an equilateral triangle with \(a=b=c,\) estimate the percent change in the area when all sides increase in length by \(p \% .\)

Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum, or a saddle point. If the Second Derivative Test is inconclusive, determine the behavior of the function at the critical points. $$f(x, y)=\tan ^{-1} x y$$

What are the conditions for a critical point of a function \(f ?\)

Let \(x, y,\) and \(z\) be nonnegative numbers with \(x+y+z=200\) a. Find the values of \(x, y,\) and \(z\) that minimize \(x^{2}+y^{2}+z^{2}\) b. Find the values of \(x, y,\) and \(z\) that minimize \(\sqrt{x^{2}+y^{2}+z^{2}}\) c. Find the values of \(x, y,\) and \(z\) that maximize \(x y z\) d. Find the values of \(x, y,\) and \(z\) that maximize \(x^{2} y^{2} z^{2}\)
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