# Thread: tangent plane to the surface

1. ## tangent plane to the surface

find the eqn of the tangent plane to the surface
z=f(x,y)=x^2 +4y^2

at the point (a,b,a^2 +4y^2)

What I don't understand is how the normal to the plane at point a,b is geven by:
n=(fx,fy,-1)

would fx not be the "rate of change with respect to x" and not the NORMAL to the plane at that point?

2. Consider the field $F(x,y,z) = x^2 + 4y^2 - z$. The surface we have is $F(x,y,z)=0$.
The normal at any point is $\nabla F(x,y,z) = \left\langle {F_x ,F_y ,F_z } \right\rangle$.
In this case it is $\left\langle {2x,8y, - 1} \right\rangle$ .

Is that your question?

3. maybe i am confused about fx, fy and fz.
I always thought it was the rate of change with respect to x and thus being the tangent to a surface?
The normal is perpendicular to the surface correct?
I have some fallacy in my understanding. Any clarification?

4. The expression $\nabla F(x,y,z) = \left\langle {F_x ,F_y ,F_z } \right\rangle$ is called the gradient of the field.
Those are partial derivatives. The gradient ‘points’ in the direction of most rapid increase from the field. That is why it is the normal of the tangent plane at any point on the surface.

5. thank you any chance you know of some sort of online visual aid/interactive program?