# Vector Analysis Homework: Finding the Unit Normal Vector at P

• manimaran1605
In summary, vector analysis is a branch of mathematics that deals with the study of vectors and their properties, such as magnitude, direction, and operations like addition and multiplication. Finding the unit normal vector at a point is important because it represents the direction perpendicular to the surface at that point, which is useful in many applications. To find the unit normal vector at a point, you first need to determine the gradient vector and then divide it by its magnitude. A unit vector has a magnitude of 1, while a normal vector is perpendicular to a surface at a given point. An example of a normal vector is a ball rolling on a curved surface, where the normal vector represents the direction of acceleration due to gravity.
manimaran1605

## Homework Statement

This picture is taken from Div, curl, grad and all that by schey, while finding the unit normal vector of the surface S (defined as w=f(x,y)) at a point P, to find the normal vector he considered a two tangent non-collinear vectors (u and v) at a point P, to find u he considered a plane passing through point P parallel to xz plane, the plane which intersects the surface S traces a curve C which contains the point P, he drawn a tangent to the curve at the point p, let the x component be ux, My question is how the z-component of u is (∂f/∂x)ux ?

## Homework Equations

No equations found

## The Attempt at a Solution

I have an idea that z-component of u is some approximation, but i havn't learn multivariable calculas a lot, so please enlighten me. Thank you

#### Attachments

• Untitled-1.jpg
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You are correct. In a small displacement from P, the linear approximation is a good one and you can write ##z(x= P + u_x) \approx z(x=P) + \frac{\partial z}{\partial x} u_x## (The Taylor expansion truncated, ignoring higher order terms). For an infintesimal displacement, this approximation becomes exact.

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## 1. What is vector analysis?

Vector analysis is a branch of mathematics that deals with the study of vectors and their properties, such as magnitude, direction, and operations like addition and multiplication. It is used to solve problems related to motion, force, and other physical phenomena.

## 2. Why is finding the unit normal vector at a point important?

The unit normal vector at a point represents the direction perpendicular to the surface at that point. This is useful in many applications, such as determining the slope of a surface, calculating the force exerted on an object by a surface, or finding the direction in which a particle will move on a curved surface.

## 3. How do you find the unit normal vector at a point?

To find the unit normal vector at a point, you first need to determine the gradient vector at that point. Then, divide the gradient vector by its magnitude to obtain the unit vector in the same direction. The resulting vector will be the unit normal vector at that point.

## 4. What is the difference between a unit vector and a normal vector?

A unit vector is a vector with a magnitude of 1, while a normal vector is a vector that is perpendicular to a surface at a given point. The unit normal vector, also known as the surface normal, is a unit vector that is perpendicular to the surface at a given point.

## 5. Can you explain the concept of a normal vector with an example?

Imagine a ball rolling on a curved surface. At any point on the surface, the normal vector will be pointing directly away from the surface, perpendicular to the tangent plane. This vector represents the direction in which the ball will accelerate due to the force of gravity. The magnitude of the normal vector will determine the steepness of the surface at that point, which will affect the ball's acceleration.

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