# Finding formula for normal vector of surface

• Somefantastik
In summary, the problem involves finding the formula for the normal to the surface of the upper half of a sphere with radius 3 at (0,0,0). The formula for the normal can be obtained by using the surface function z = f(x,y) and the formula for the unit normal vector. The unit normal vector can be simplified by using implicit differentiation. However, the resulting expression may appear complex, but this is not necessarily incorrect. Finally, it is suggested that there may be an easier way to find the unit normal vector for a sphere.
Somefantastik

## Homework Statement

S is surface of upper half of sphere (rad 3) at (0,0,0). Find the formula for n (in Cartesian, spherical, and cylindrical coord syst), then evaluate

$$\int\int\textbf{F}\cdot\textbf{n}dA$$

Where F(x,y,z) = k

## The Attempt at a Solution

in a previous problem, if I let z = f(x,y) be the surface and take

$$\psi(x,y,z) = f(x,y) - z$$

Since $$\nabla\psi$$ is orthogonal to the surface on which $$\psi$$ is constant, then I can use this to get the formula for the normal to the surface.

Problem is, I'm stuck trying to figure out what the formula for the surface is to start with.

A sphere with radius 3 is just x2+y2+z2 = 3

so

$$z = 3 - \sqrt{x^{2}+y^{2}}$$

where 0< x,y < 3

now

$$\textbf{n} = \frac{\nabla\psi}{\left\|\nabla\psi\right\|}$$

Am I on the right track? The unit normal vector field gets pretty ugly after this step :-/

It can't be that bad, can it?

BTW, using implicit differentiation then solving for the derivative is often much simpler that solving for the variable and differentiating that.

By the way, surely you know enough about spheres to have an easy way of finding the unit normal?

I must have done something wrong because I've gotten something pretty heinous.

$$\nabla\psi = -x(x^{2} + y^{2})^{-1/2}\textbf{i} - y(x^{2} + y^{2})^{-1/2}\textbf{j} - \textbf{k}$$

$$\left\|\nabla\psi \right\| = (x^{2} + y^{2}) + 1$$

I believe that is wrong, but what's so "heinous" about that expression? The correct one has the same sort of appearance.

"By the way, surely you know enough about spheres to have an easy way of finding the unit normal? "

Obviously not, can you expound a little?

## What is a normal vector of a surface?

A normal vector of a surface is a vector that is perpendicular to the surface at a given point. It represents the direction in which the surface is pointing outward.

## Why is it important to find the formula for the normal vector of a surface?

Knowing the formula for the normal vector of a surface allows us to calculate the direction and magnitude of the vector at any given point on the surface. This information is crucial in many applications, such as determining the angle of incidence and reflection of light on a surface, or calculating the force exerted by a surface on an object.

## How can the formula for the normal vector of a surface be derived?

The formula for the normal vector of a surface can be derived using vector calculus and the gradient operator. The gradient of a function represents the direction and magnitude of the steepest ascent of the function at a given point. By taking the gradient of the surface's equation, we can obtain the formula for the normal vector.

## Are there any special cases when finding the formula for the normal vector of a surface?

Yes, there are some special cases where the formula for the normal vector may be different. For example, if the surface is a plane, the normal vector can simply be calculated by taking the cross product of two non-parallel vectors on the plane. Another special case is when the surface is defined implicitly by an equation, in which case the formula for the normal vector can be derived using implicit differentiation.

## What are some real-world applications of the normal vector of a surface?

The normal vector of a surface has a wide range of applications in various fields. In engineering, it is used to calculate the stress and strain on different parts of a structure. In physics, it is used to calculate the electric and magnetic fields around charged objects. In computer graphics, it is used to determine the shading and lighting of 3D objects. Additionally, the normal vector is also used in navigation and robotics for determining the orientation and direction of movement.

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