Show that the volume of a solid ball of radius R in 3-dimensions is Hint: use...

 

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Show that the volume of a solid ball of radius R in 3-dimensions is Hint: use the following procedure 1. use the spherical polar coordinates system to determine the vectors tangent to surface of the sphere. Suppose that the north pole corresponds to the point = 0. 2, compute the metric tensor g in this coordinate system. 3. determine the volume element by computing the scalar triple product of the tangent vectors. 4. compare the volume element with det (g), where det(g) is the absolute value of the determinant of the metric tensor (when written as a matrix). 5. integrate volume element of the sphere over the northern hemisphere and then double the value of the integral to get the volume of the entire sphere. 6. integrate volume element of the sphere over the entire sphere and compare your answer with that obtained by doubling the volume of the northern hemisphere. (Should these answers coincide?) Show that the volume of a solid ball of radius R in 3-dimensions is Hint: use the following procedure 1. use the spherical polar coordinates system to determine the vectors tangent to surface of the sphere. Suppose that the north pole corresponds to the point = 0. 2, compute the metric tensor g in this coordinate system. 3. determine the volume element by computing the scalar triple product of the tangent vectors. 4. compare the volume element with det (g), where det(g) is the absolute value of the determinant of the metric tensor (when written as a matrix). 5. integrate volume element of the sphere over the northern hemisphere and then double the value of the integral to get the volume of the entire sphere. 6. integrate volume element of the sphere over the entire sphere and compare your answer with that obtained by doubling the volume of the northern hemisphere. (Should these answers coincide?)

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To show that the volume of a solid ball of radius R in 3 dimensions is 43R3 we can follow the steps outlined in the hint Step 1 Use spherical polar co... View the full answer