Determine the tangent planes of the level surface given by x^2+y^2+x^2 = 1 at the points (x,y,0) and show that they are all parallel to the z-axis.
here is my working thus far:
f(x,y,z) = x^2+y^2+z^2-1
then the tangent plane equation is:
(∂f/dx)|(x,y,0)(x-x) + (∂f/dy)|(x,y,0)(y-y) + (∂f/dz)|(x,y,0)(z-0) = 0
the first two terms go to zero since (x-x = y-y = 0), and the last term leaves 0z=0 (no information!)
Mathematically, where is my problem?
the question is asking me for the tangent planes though, so there lies my problem.
As shown in my original post, I get the erroneous answer 0z=0! So how do I fix this?
Also, when you say there is no z-coordinate, I cannot see it being parallel to the z-axis, but rather perpendicular to the x-axis. (Imagine x-axis to the right, y-axis out of the page, and z-axis up. Now the xy plane is perpendicular to the z-axis!)
Think of
2x0(x-x0) + 2y0(y-y0) = 0
as a line in the x-y plane that is then extended in the pos and neg z directions.
Your original surface is a sphere- the equator is in the x-y plane
The tangent planes at the equator are vertical planes whose equations are given by:
2x0(x-x0) + 2y0(y-y0) = 0
See attachment
Remember that to find the equation that describes a plane you need to have a point on a plane and the vector to which the plane is normal (perpendicular) to. When we compute the gradient at (x,y,0) we are finding the normal vector to the tangent plane at that point. The normal vector works out to be (2x,2y,0), notice that this vector has no k component, so it must entirely lie in the xy-plane. Therefore, the tangent plane is perpendicular to the xy-plane. But the z-axis is perpendicular to the xy-plane. Now since the tangent plane and the z-axis are both perpendicular to the xy-plane it means they must be parallel with eachother.
ok, I think i'm beginning to understand.
before I was confused by the notation (I'm guessing that P should be more explicitely explained as (x0,y0,0)
Let me explain what I understand so far to see if I'm on the right track..
so, to find my tangent planes, I'm using the tangent plane equation,
z-z0 = (∂f/dx)|(x0,y0,0)(x-x0) + (∂f/dy)|(x0,y0,0)(y-y0)
=> z = 2x0(x-x0) + 2y0(y-y0)
Now to prove these are parallel to the z-axis, I'm going to show that the normal to the tangent plane has no z-component, or if I wanted to go a little bit further, that the normal vector n dot produced with the z-axis vector, is 0 (i.e the normal of the tangent plane is perpendicular to the z-axis, thus the tangent plane itself is parallel to the z-axis). Hopefully that makes sense!
So now..
n = <2x 2y 2z>
evaluated at (x0,y0,0), n = <2x0 2y0 0>
no z-component, so parallel to z-axis.
Extra step...
z-axis vector = k = <0 0 1>
n dot k = <0 0 0> = 0
i.e. normal of tangent plane is perp to z-axis, therefore tangent plane is parallel.
Please tell me I'm correct!
Yes, this is what we have been saying.
But you got the equation of the tangent plane wrong. This is because the tangent plane equation that you used only works when . But is given implicitly not explicitly. So be careful. You can solve for z then use tangent plane equation but those steps are not necessary.
You ned to use the chain rule to find . The tangent plane is given by . However, this formula would only work when is differenciable at . The function it not differenciable on the set . Think of it this way. You have the function (in a single variable) . You can draw the tangent line at , that line has equation , however you cannot find the line by using the formula because this formula only works when the function is differenciable at the point. Notice that as for the slope of tangent line shoots away , so the tangent slope does not exist. Likewise, with on the circle . For all these points as you start drawing the tangent plane it becomes vertical and so the function fails to be differenciable. This is way you need to solve this problem by leaving the surface in the form and applying the gradient to this function as you did in the beginning.
I'm confused at this point!
Are you saying I should be using the tangent plane equation:
(∂f/dx)|(x,y,0)(x-x0) + (∂f/dy)|(x0,y0,0)(y-y0) + (∂f/dz)|(x0,y0,0)(z-z0) = 0
If I do this though, ∂f/dz = 2z, then when evaluated at (x0,y0,0), this goes to zero and I am left with:
2x0(x-x0) + 2y0(y-y0) = 0
Are you saying this is my tangent plane equation?