Calc 3 Lecture NotesSection 12.6Page 1 of 8
Section 12.6: The Gradient and Directional Derivatives
Big idea:Derivatives in directions other that the x and y directions can be computed using a linear combination of partial derivatives. The form for these directional derivatives can be represented compactly using a new operator called the gradient operator.
Big skill: You should be able to compute the gradient of a function, and use it to compute a directional derivative and the tangent plane to an implicitly defined surface.
Definition 6.1: The Directional Derivative
The directional derivative of f(x, y) at the point (a, b) and in the direction of is given by:
Theorem 6.1: The Directional Derivativein Terms of Partial Derivatives
If f(x, y) is differentiable at the point (a, b) and is any unit vector, then we can write:
Practice:
- For the function , find in the direction of the vector using the limit definition and Theorem 6.1.
- The image below shows the unit vector superimposed on level curves from
z = 1.2 to z = 3.0 in steps of z = 0.2 of the surface near the point (1, 2, 1.4). Use the image to compute an approximation of along u.
Definition 6.2: The Gradient Vector
The gradient of f(x, y) is the vector-valued function
,
provided both partial derivatives exist.
Theorem 6.2: The Directional Derivative in Terms of the Gradient
If f(x, y) is differentiable and is any unit vector, then:
Practice:
- Let
- Compute
- Compute
- Compute , where and .
Now that we know how to compute the rate of change along any given arbitrary direction, we can ask what direction maximizes the rate of change? The answer can be obtained from the alternate definition of the dot product:
Thus, the maximum change occurs when the angle between u and the gradient is 0 or 180. That is to say, the rate of change of the function is greatest along it gradient.
Theorem 6.3: Properties of the Gradient
If f(x, y) is differentiable at the point (a, b), then:
(i).The maximum rate of change of f at (a, b) is , occurring in the direction of the gradient.
(ii).The minimum rate of change of f at (a, b) is -, occurring in the direction opposite of the gradient.
(iii).The rate of change of f at (a, b) is 0 in directions orthogonal to the gradient.
(iv).The gradient is orthogonal to the level curve f(x, y) = f(a, b).
Practice:
- Find the values for and directions of the maximum and minimum rates of change of the function at the point (1, 2). Notice how the unit vector in the gradient direction is perpendicular to the level curves.
- Find the values for and directions of the maximum and minimum rates of change of the function at the point (0, 1).
- A man stands at the point on a hill whose elevation is given by . In what direction should he begin to walk in order to climb the hill most rapidly? Assuming he walks in this direction, what will be his rate of ascent initially? Trace out an approximately optimal path on the level curves shown below for the path of steepest ascent.
Definition 6.3: The Gradient Vector for a 3 Variable Function
The directional derivative of f(x, y, z) at the point (a, b, c) and in the direction of is given by:
The gradient of f(x, y, z) is the vector-valued function
.
Definition 6.4: The Directional Derivative in Terms of the Gradient Vector
Practice:
- If the temperature T at a point (x, y, z) in a solid is given by , find the direction from the point (2, 0, 50) in which the temperature increases most rapidly.
Theorem 6.5: Tangent Plane to an Implicit Equation of Three Variables.
If the point lies on the surface defined by , and the three partial derivatives , , and all exist at that point, then the vector is normal to the surface at that point, and the tangent plane is given by the equation
The normal line through the surface at this point is given by:
Practice:
- Find equations of the tangent plane and normal line to the surface defined implicitly by:
at the point (1, 2, 3).
- Find the equation of the tangent plane to the surface by: at the point
(, , 0).