Question
Find a linear transformation that maps the unit sphere $x^2+y^2+z^2=1$ to the ellipsoid $x^2+\frac{1}{4} y^2+\frac{1}{16} z^2=1$
Step 1
The goal is to find a linear transformation \( T \) that maps every point on the unit sphere \( x^2 + y^2 + z^2 = 1 \) to a corresponding point on the ellipsoid \( x^2 + \frac{1}{4} y^2 + \frac{1}{16} z^2 = 1 \). Show more…
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Let T : R^3 -> R^3 be the linear transformation determined by the matrix A = [a 0 0; 0 b 0; 0 0 c] where a, b, and c are positive numbers. Let S be the unit sphere, whose bounding surface has the equation x_1^2 + x_2^2 + x_3^2 = 1. (a) Show that T(S) is bounded by the ellipsoid with the equation x_1^2/a^2 + x_2^2/b^2 + x_3^2/c^2 = 1. (b) Use the fact that the volume of a sphere with radius 1 is 4π/3 to determine the volume of an ellipsoid with axes with lengths 2a, 2b, and 2c.
Let T : R³ → R³ be the linear transformation determined by the matrix A = [[a, 0, 0], [0, b, 0], [0, 0, c]], where a, b and c are positive real numbers. Let B be the unit sphere defined by x₁² + x₂² + x₃² = 1. (a) Show that the image T(B) is the ellipsoid with the equation x₁²/a² + x₂²/b² + x₃²/c² = 1. (b) Use determinants and the fact that the volume of the unit sphere is 4/3π to determine the volume formula for the ellipsoid in part (a).
Convert the equation into spherical coordinates. $$x^{2}+y^{2}+(z-1)^{2}=1$$
Multiple Integrals
Spherical Coordinates
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