Computing the second fundamental form - can you spot the error?

In summary, the conversation revolves around computing the second fundamental form for a surface. The coefficients for IIuu, IIuv, and IIvv should be 2, 0, and -2 respectively. However, there seems to be an error in computing IIx and IIy, which should be the same as IIz. The formula used for calculating the second fundamental form was also mentioned.
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Homework Statement
Compute the second fundamental form of the surface.
Relevant Equations
Standard differential geometry formulas for the second fundamental form with Christoffel symbols.
Hi all. We are computing the second fundamental form corresponding to a surface.

The correct answer should give the coefficients 2, 0, and -2 for IIuu, IIuv, IIvv

We have only been able to compute IIz, and not IIx, IIy where we have functions instead of 2, 0 and -2.

We are 99% certain that everything up to the da component in part (e.) is correct. It is just in the last computations where an apparent errors are. Can you see where the error is in part g?
Screen Shot 2020-12-20 at 8.06.39 PM.png


Screen Shot 2020-12-20 at 8.06.44 PM.png

Screen Shot 2020-12-20 at 8.06.51 PM.png

Screen Shot 2020-12-20 at 8.06.55 PM.png


Screen Shot 2020-12-20 at 8.06.59 PM.png

Screen Shot 2020-12-20 at 8.07.03 PM.png


In part g, the II's corresponding to x and y should be the same as z. It seems like something is missing, perhaps. The second fundamental form were calculated using the following formula.
Screen Shot 2020-12-20 at 8.22.34 PM.png


Thank you!
 
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I am replying to this thread so it no longer shows up on the "unanswered threads" list
 

1. What is the second fundamental form in computing?

The second fundamental form is a mathematical concept used in differential geometry to measure the curvature of a surface. It is a tensor that describes how the surface curves in different directions.

2. Why is computing the second fundamental form important?

Computing the second fundamental form is important because it helps us understand the geometric properties of a surface, such as its curvature and shape. It is also used in various applications, such as computer graphics and physics.

3. How is the second fundamental form calculated?

The second fundamental form is calculated by taking the derivative of the surface's normal vector with respect to the surface's tangent vectors. It can also be calculated using the Gaussian curvature and mean curvature of the surface.

4. What are some common errors when computing the second fundamental form?

Some common errors when computing the second fundamental form include using incorrect formulas, not properly taking into account the orientation of the surface, and making mistakes in the calculations.

5. How can I spot errors when computing the second fundamental form?

To spot errors when computing the second fundamental form, it is important to double check all formulas and calculations, pay attention to the orientation of the surface, and compare the results to known values or previous calculations. It is also helpful to have a good understanding of the underlying concepts and equations involved.

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