Question

Differential Geometry (Mixed Use of Vector Calculus & Linear Algebra)

1A. Prove that if p=(x,y) is in the set where y<x and if r=distance from p to the line y=x then the ball about p of radius r does not intersect with the line y=x.

1B. Prove that the set where y<c is an open set.

Answer 1

1A. Suppose de notes the distance between the points and .  

r = distance from the point p to the line y=x .

There exist such that .

Let be any point on the line .

Now ,

The point does not lie inside the ball centered at p and radius r .

As be arbitrary point on the line . Hence the line does not intersect the ball centered at p and radius r .

1B. Let .

We will prove that the set B is open .

Let , then .

.

If we choose

Then the ball centered at and radius entirely contained in .  

(x,y) is an interior point of B .

As (x,y ) is arbitrary so every point of B is an interior point of B .

Hence the set B is open .

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If you have doubt or need more clarification at any step please comment.