What is the gradient of your walk? a. ∼0.003 b. ∼0.05 c. ∼0.5 d. ∼1

Name: Problem 27, Chapter 2: Laboratory Manual for Introductory Geology
Uploaded: 2020-06-02T04:38:13-08:00
Duration: 5 min 20 s
Channel: Megan Mcfarland
Description: What is the gradient of your walk? a. ∼0.003 b. ∼0.05 c. ∼0.5 d. ∼1

What is the gradient of your walk? a. ∼0.003 b. ∼0.05 c....

Key Concepts

Gradient

In mathematics and physics, the gradient refers to the rate of change of a quantity, typically representing how much one variable changes in relation to another. When applied to a physical context like a walk or a path, it quantifies the steepness of the slope by comparing the vertical rise to the horizontal distance traveled.

Slope

Slope is a fundamental concept in geometry and calculus that describes the inclination or decline of a line or surface. It is computed as the ratio of the vertical change (rise) to the horizontal change (run), offering a clear measure of the steepness encountered in scenarios such as traversing varied terrain.

Transcript

00:01 In this problem, we are given the shown information, and first ask to provide the gradient of f.

00:06 To do this, we need the partial of f with respect to x, with respect to y, with respect to z.

00:12 And so first, we're going to find the partial with respect to x.

00:18 And that means we're going to treat y and z as constants.

00:21 And so when we do this, we get 2x, y, z, minus y, z cubed.

00:30 And now we're going to take the partial with respect to y, and that is going to be equal to the following x and z are our constants and so we're going to get x squared z minus x z cubed and now we're going to find the partial with respect to z so x and y are our constants and that is going to be equal to x squared y minus x y times three z squared which is equal to x squared squared y minus 3 x y z squared now we're going to write this in a vector in order to find our gradient and so we're going to get the following 2 x y z minus y z cubed x squared z minus x z cubed x squared z minus x z cubed x squared y x squared y minus x squared y minus three x y z squared and that is our answer to part a in part b we are told to plug in the point p into what we just found and so when we do this we get the following two times two times negative one times one minus negative one times z cubed which is just one comma two squared is four times one minus one excuse me times minus two times one cubed which is just one comma two squared is four times negative one minus three times two times negative one times z squared one squared which is just one.

02:57 Okay now we have to simplify this and when we do we get two times two is four times negative 1 times 1, so that's 4 times negative 1, that's negative 1, minus negative 1 times 1.

03:11 Or we're going to get negative 4 plus 1, that's negative 3, and then we get 4 times 1, minus 2 times 1, 4 minus 2 is 2, and then we get 4 times negative 1, that's negative 4, minus 3 times 2 times 2, that's going to be minus negative 6, so we get negative 4 plus 6, that's 2.

03:42 And that is our answer to part b.

03:46 Now in part c, we are told to provide the rate of change of f at point p in the direction of the vector that they give us.

03:59 Now we need to make sure that the length of the vector they give us is 1.

04:02 And so to do that, we're going to take the square root of each of its components squared...

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