# Vector calculus finding the surface normal

• frixis
In summary, the conversation is about the book "div, grad, curl and all that" and the confusion surrounding the calculation of the components of the tangent vectors u and v. The person is asking for an explanation of how u is calculated and how it relates to the function tangent in the x direction, as well as how v is related to the function tangent in the y direction. They mention the orthonormal vectors i, j, and k and their corresponding units in relation to the x, y, and z components.
frixis
okay i was reading the book "div, grad cur and all that"
i've just started for my exam. but i got stuck in the beginning. I'm attaching the page because i really don't get it .
basically pages 14 and 15. i don't get how they calculate the components of u and v.
if someone explains u i'll get v obviously.
thanks

#### Attachments

• dontget.JPG
26 KB · Views: 542
frixis said:
okay i was reading the book "div, grad cur and all that"
i've just started for my exam. but i got stuck in the beginning. I'm attaching the page because i really don't get it .
basically pages 14 and 15. i don't get how they calculate the components of u and v.
if someone explains u i'll get v obviously.
thanks

U is calculated as a tangent vector. Consider how the rate of change impacts the tangent vector. Think about how a steep surface or how a flat surface would impact the definition of u. Its probably best if visualize a normal one dimensional function and the normal tangent vector to that function.

If you relate u vector to the function tangent in the x direction and v vector to that in the y direction you have two vectors that are orthogonal to each other and represent the best information to calculate a normal since the normal is u x v.

On a steep surface we would have a normal that would have a smaller z component than if the surface were flat on the x,y plane.

In terms of calculation they are converting the x and z components which is df/dx * ux or whatever it says in the book by writing it in terms of its components which are in terms of the orthonormal vectors i j and k which correspond to the unit vectors of the x y and z axis respectively. Just reflect on what units everything is in by looking at the diagram and see how the units for x and z correspond to units or i and k respectively.

## 1. What is the surface normal in vector calculus?

The surface normal in vector calculus is a unit vector that is perpendicular to a given surface at a specific point. It is used to determine the direction of the surface's normal force and is an important concept in fields such as physics, engineering, and computer graphics.

## 2. How is the surface normal calculated in vector calculus?

The surface normal is calculated by taking the cross product of two tangent vectors on the surface. These tangent vectors are typically the partial derivatives of the surface's equation with respect to its two independent variables.

## 3. What is the significance of finding the surface normal in vector calculus?

Finding the surface normal allows us to determine the direction of the normal force acting on the surface. This information is crucial in understanding the behavior of objects on or interacting with the surface, such as the path of a projectile or the force of friction.

## 4. Can the surface normal change at different points on the same surface?

Yes, the surface normal can change at different points on the same surface. This is because the tangent vectors used to calculate the surface normal may vary depending on the location on the surface.

## 5. How is the surface normal used in practical applications?

The surface normal is used in many practical applications, including 3D modeling and computer graphics, fluid dynamics, and mechanics. It is also used in fields such as computer vision and robotics, where the understanding of surface orientation is crucial for accurate detection and manipulation of objects.

Replies
1
Views
1K
Replies
6
Views
2K
Replies
5
Views
1K
Replies
1
Views
1K
Replies
3
Views
2K
Replies
4
Views
2K
Replies
6
Views
4K
Replies
6
Views
2K
Replies
4
Views
1K
Replies
2
Views
2K