# Normal Vector for (x,y,z) Surface of f(x,y)

• EV33
In summary, the formula (x,y,z)+t(f_{x},f_{y},-1) is used to find the normal vector for a point (x,y,z) on a surface of some function f(x,y). It is obtained by taking the scalar product of the tangent vector with (f_x,f_y,-1), which is always orthogonal to the surface. This formula can also be used to find the normal vector for a level surface of a function F(x, y, z)= f(x,y)- z.
EV33

## Homework Statement

My question is if this is the formula for a normal vector for the point (x,y,z) of a surface of some function f(x,y).

(x,y,z)+t(f$$_{x}$$,f$$_{y}$$,-1)

My teacher used it in class and I just wanted to know if it is what I think it is.

## The Attempt at a Solution

Thank you.

Those superscripts should be subscripts.

Your surface is (x,y,f(x,y)). Take a path in (x,y) plane: (x(s),y(s)) - s is the parameter along the path. The path on your surface is (x(s),y(s),f(x(s),y(s))). Differentiate at s=0, denote the derivative by a dot. You get for the tangent vector:

$$({\dot x},{\dot y},{f_x{\dot x}+f_y{\dot y})$$

Take the scalar product with $$(f_x,f_y,-1)$$ - you will get automatically 0. So you get a vector that is orthogonal to the surface. You may like to normalize its length - multiply by an appropriate number t, and then move it to the point on the surface (x,y,z=f(x,y)). The result is as in your formula.

If z= f(x,y), then we can think of the surface as a "level surface" for some function F(x, y, z)= f(x,y)- z. The gradient $\nabla F= f_x\vec{i}+ f_y\vec{j}- \vec{k}$ is always normal to a level surface and so is a normal vector for that surface.

What you have is the vector equation for a line perpendicular to the surface. For fixed t it gives a normal vector.

## 1. What is a normal vector for a surface in (x,y,z) coordinates?

A normal vector for a surface in (x,y,z) coordinates is a vector that is perpendicular to the surface at a given point. It is used to determine the orientation of the surface and is an important concept in vector calculus.

## 2. How is the normal vector calculated for a given point on a surface?

The normal vector for a given point on a surface can be calculated by finding the partial derivatives of the surface with respect to x and y, and using them to construct a vector that is perpendicular to the surface at that point.

## 3. Can a normal vector change at different points on a surface?

Yes, a normal vector can change at different points on a surface. This is because the orientation of the surface may change, and therefore the direction of the perpendicular vector will also change.

## 4. How is the normal vector used in surface integrals?

The normal vector is used in surface integrals to calculate the flux, or the flow of a vector field through a surface. It is also used to determine the direction of the surface element in the integration process.

## 5. What is the significance of the normal vector in gradient and directional derivative calculations?

The normal vector is important in gradient and directional derivative calculations because it represents the direction of steepest ascent or descent at a given point on the surface. This allows for the calculation of the rate of change of a function in a specific direction.

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