"Hence the partial derivatives ru and rv at P are tangential

In summary, the equation for r tilde prime in the attached image implies that the partial derivatives ru and rv at P are tangential to the surface S at P. This can be understood by considering how varying u and v independently will result in infinitesimal motion that is tangent to the surface, and taking the cross product of the two partial derivatives will give a vector normal to the tangent plane at that point. This is similar to finding the tangent line of a function in one dimension, but in this case the partial derivatives generate a tangent plane for the surface S.
  • #1
OmegaKV
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I've been looking at the equation for r tilde prime in the image I attached below, but I cannot understand why it is that they say "Hence, the partial derivatives ru and rv at P are tangential to S at P".

g0gYbLr.jpg


How does that equation imply that ru and rv are tangential to P?
 
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  • #2
When you parametize a surface and have some position function ## r=r(u,v) ## that defines the surface, you can vary u and v independently and any new value of u and/or v (e.g. ## u_1 ## and ## v_1 ##) will give a location ## r=r(u_1,v_1) ## that is also on the surface. As a result, if you allow just u to vary infinitesimally (keeping v constant), you will move along the surface and your direction of motion will be tangent to the (two-dimensional) surface. Likewise, infinitesimal variations in v (keeping u constant) will also cause you to move tangent to the surface, i.e. the motion is somewhere in the plane that is tangent to the surface at that point. Normally the motion caused by varying u and the motion caused by varying v are not colinear. The two partial derivatives (by taking a vector cross product) will give you a vector normal to the tangent plane at that point(i.e. perpendicular to the surface).
 
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  • #3
Is as in one dimension where the angular coefficient of the tangent line of a function ##f(x)## is the derivative in this point. In this case you have that the surface ##S## is a function of two variables ##f(x,y)## so the partial derivatives generate a plane (not a line!) that is the tangent plane at ##P## ... I simplify a lot with the hope to clarify the concept ...
 

1. What is the meaning of "partial derivatives"?

Partial derivatives refer to the rate of change of a function with respect to one of its variables, while holding all other variables constant. In other words, it measures how a function changes when only one of its variables is changed.

2. How are partial derivatives represented mathematically?

Partial derivatives are represented using the symbol ∂ (pronounced "del"), followed by the variable with respect to which the derivative is being taken. For example, ∂f/∂x represents the partial derivative of f with respect to x.

3. What is the significance of "tangential" in this context?

In this context, "tangential" means that the partial derivatives at point P are parallel to the tangent plane of the function at that point. This helps us understand the direction and rate of change of the function at that specific point.

4. How do the partial derivatives at point P relate to the overall function?

The partial derivatives at point P provide information about the behavior of the function in the vicinity of that point. They can help us determine the critical points, local extrema, and the shape of the function in that region.

5. Why are partial derivatives important in scientific research?

Partial derivatives are important in scientific research as they allow us to analyze the behavior of complex functions with multiple variables. They are essential in fields such as physics, economics, and engineering, where understanding the rate of change of a system is crucial for making predictions and solving problems.

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