# Lecture 12: Gradient, directional derivative, and tangent plane

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Lesson Summary:

In this lesson, you will learn about the gradient vector, directional derivative, and tangent planes. The gradient vector is a vector formed by putting together all of the partial derivatives that gives you a vector field. The directional derivative is the dot product between the gradient and the velocity vector, which is also perpendicular to the level surface. The tangent plane is the best way to approximate the function at a given point, and it has the same normal vector as the surface. With this understanding, you can find the tangent plane to any surface given an equation and a point on the surface.

Lesson Description:

Learn about the gradient vector, directional derivative, and more on the tangent plane.

Denis Auroux. 18.02 Multivariable Calculus, Fall 2007. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (Accessed March 15, 2009). License: Creative Commons Attribution-Noncommercial-Share Alike.

• What is the gradient vector and how do you find it?
• What does the gradient vector mean, what does it measure, and what can you do with it?
• How do you find the normal vector to a plane?
• How do you find the level curve of a function?
• What is the proof that the gradient is perpendicular to the level surface for a function?
• How do you find the tangent plane to a surface using a gradient?
• What are directional derivatives and how do you find them?
• What do directional derivatives mean geometrically?
• How can you determine the direction that a gradient vector is pointing?
• #### Staff Review

• Currently 4.0/5 Stars.
The gradient vector is discussed and explained in depth after being introduced in the previous lecture. This idea is very important to the idea of vector fields, which are discussed later in this course. Directional derivatives are also explained and shown very using pictures and problems.