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 .
2) z = (x^2) + (y^2) .
3) z = y .
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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>.
More formally, the gradient of F(x,y,z) = z - 3 = 0 will be normal to the surface,
and grad F =
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>.