8) Answer the question shown in the image 8. Consider the surface S of equation z2-2x? + y²-x= 0 and letP (2, 3, 1) be a point at S. Suppose z is an implicit functionof x and y in a neighborhood of point P. Consider thefollowing statements:II. The vector Vz (P) is unitary.Is this true or false?

Answered: 8. Consider the surface S of equation…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

8) Answer the question shown in the image

8. Consider the surface S of equation z2-2x? + y²-x= 0 and let
P (2, 3, 1) be a point at S. Suppose z is an implicit function
of x and y in a neighborhood of point P. Consider the
following statements:
II. The vector Vz (P) is unitary.
Is this true or false?

Expert Solution

Step 1

In this question, concept of gradient is used.

Gradient

A scalar field's gradient may be a vector field whose magnitude is that the rate of change and which points within the direction of the scalar field's greatest rate of rise. If the vector is resolved, the components of the vector describe the scalar field's rate of change with regard to each directional component.

Gradient of field $f (x, y, z) = < \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} >$