Vector Calculus :) please help
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Vector Calculus :) please help

[From: ] [author: ] [Date: 13-02-16] [Hit: ]
3) z = y .-1.) z = 3 is a plane parallel to the xy-plane, and so the unit normal is .More formally, the gradient of F(x,......
For each of the following surfaces, find a mathematical expression for the unit normal.

1) z = 3 .
2) z = (x^2) + (y^2) .
3) z = y .

-
1.) z = 3 is a plane parallel to the xy-plane, and so the unit normal is <0,0,1>.

More formally, the gradient of F(x,y,z) = z - 3 = 0 will be normal to the surface,

and grad F = = <0,0,1>, where the derivatives are partials.

2.) With F(x,y,z) = x^2 + y^2 - z = 0, we have grad F = <2x, 2y, -1>, and so the

unit normal will be

(1/sqrt((2x)^2 + (2y)^2 + (-1)^2) * <2x, 2y, -1> = (1/sqrt(4*z + 1))*<2x, 2y, -1>.

3.) With F(x,y,z) = y - z = 0, we have grad F = <0,1,-1>, and thus the unit normal

will be (1/sqrt(2))*<0,1,-1>.
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