Geometric Meaning of Parital Derivative (specific problem)

Question

I have an article in my handouts which I doubt is wrong. Article is following:

"Geometric Meaning of Partial Derivatives
Suppose $z = f ( x , y )$ is a function of two variables.The graph of $f$ is a surface. Let $P$ be a point on the graph with coordinates $( x_0, y_0 , f(x_0 , y_0 ))$. If a point starting from $P$, changes its position on the surface such that $y$ remains constant, then the locus of this point is the curve of intersection of $z= f (x, y )$ and $y = $constant. On this curve, $\frac{\partial z}{\partial x}$ is derivative of $z = f (x , y)$ with respect to $x$ with $y$ constant......"

Now this article says locus of this point (means locus of one single point $(x_0, y_0)$) is curve of intersection of surface $f(x, y)$ and plane $y = y_0$
But to my knowledge locus of single point is nothing but a circle so

1. Either every curve got by intersection of a surface and a plane is circle (which is a straight lie)
   OR
   

2. This article is making a mistake here
   OR
   

3. There is something else which I don't know.
   
   

Answer 1

In mathematics the term "Locus" (which is Latin for point or position) means a set of points satisfying some criterion.

In your case, the meaning is as follows. Take a two-dimensional curved surface, where the height $z$ is a function of $x$ and $y$, in other words: $z = f(x, y)$. Take an arbitrary point $P = (x_0, y_0, z_0)$ on this surface. Now look what happens if we vary $x$ around $x_0$, while keeping $y$ fixed. This gives you "the locus": a set of points $(x, y_0, f(x, y_0))$, which is a one-dimensional curved line on your surface. We can take the derivative of this curved line in the standard way. And that is how we define the partial derivative with respect to $x$.