Question

For every \(\mathbf{t} \in \mathcal{T}_{P}(M)\) and every constant function \(c \in F^{\infty}(P)\), show that \(\mathbf{t}(c)=0\). Hint: Use both parts of Definition \(28.2 .3\) on the two functions \(f=c\) and \(g=1\).

Short answer

From Definition 28.2.3, and given that the function c is constant, its derivative with respect to any tangent vector \( \mathbf{t} \) on the manifold is zero. Hence, we proved that \(\mathbf{t}(c)=0\) for all \(\mathbf{t} \in \mathcal{T}_{P}(M)\) and \(c \in F^{\infty}(P)\).

Step-by-step solution

Understanding Definition 28.2.3

First, look up Definition 28.2.3 from the book. This definition likely describes how to take the derivative of a function with respect to a tangent vector to a manifold (M). Break it down in own terms to understand it.

Applying the definition to a constant function

Next, apply this definition to a constant function c, with f = c and g = 1. Because c is a constant function, it has the same value at every point on our manifold (M). Thus, its differential (which measures how much the function changes as we 'move' along the manifold) is zero everywhere.

Calculation

Then, calculate the derivative of c with respect to the tangent vector: according to Definition 28.2.3, depending on the specific form of the definition, this might involve taking the limit of a difference quotient of the function values at two points in M, or calculating a line integral along a curve in M. Regardless of the details, because c is constant and its differential is zero, we should find that the derivative of c with respect to the tangent vector is zero.

Conclusion

Lastly, based on the calculations in Step 3, conclude that the required statement is true: for every tangent vector and every constant function, the derivative of the function with respect to the tangent vector is zero, that is \(\mathbf{t}(c)=0\). This matches our intuition that the derivative of a constant function is always zero.

Key concepts

Tangent Vector

In calculus, a tangent vector is a fundamental concept that not only aids in understanding curves but also higher-dimensional surfaces known as manifolds. Think of a tangent vector as an arrow touching a curve at a single point, gliding along the surface without digging into or lifting off from it. This vector indicates the direction in which the curve proceeds at that instance.

For a curve, the tangent vector points in the direction of the curve's local movement. Imagine a car driving along a hilly road; the direction it's currently heading is analogous to the tangent vector. In the context of manifolds, which can be curves, surfaces, or even more complex structures, a tangent vector at a point provides the notion of an immediate direction of travel within the manifold at that point.

The exercise presented highlights an essential characteristic of tangent vectors: their interaction with functions on the manifold, particularly constant functions. Since a constant function does not change, no matter which direction we 'travel' along the manifold, the tangent vector always zeroes out when applied to it.

Differential of a Function

The differential of a function is a term that pops up frequently in calculus, specifically when discussing changes and rates. It helps to imagine it as a mathematical tool that lets you zoom in on how a function alters as you nudge your position just a smidge. It's like looking at a hiking trail map to see how the elevation changes as you take a small step in any direction.

In formal terms, the differential captures the best linear approximation of how the function changes near a given point, and in many cases can be represented by the derivative of the function. If we take it back to our manifold, the differential of a function there would be akin to measuring how the function's output shifts as we explore different paths on the manifold, represented by different tangent vectors.

When we apply this to a constant function which has the same value everywhere on the manifold, the differential, understandably, comes out to zero. This insight is used in the exercise to show that when you pass a constant function through the gears of the differential machinery, the output is zero, aligning with the intuitive understanding that constants don't change, and hence, their rate of change or differential is nil.

Manifold

A manifold is a shape that can seem overwhelmingly complex but can actually be thought of quite simply. A manifold is a space where, if we zoom in closely enough, everything looks flat, like a small piece of paper. It might be a curve (1-dimensional), a surface like a globe (2-dimensional), or something that stretches out in even more directions, just like the universe does. The crucial aspect is that, locally, it resembles flat, familiar Euclidean space.

Manifolds are incredibly important in higher-dimension calculus because they are the playgrounds where functions live and where tangent vectors have races. They generalize concepts such as curves and surfaces to any dimension, permitting us to study a plethora of spaces using the calculus we know and love. Furthermore, in this exercise, the manifold serves as the domain for the functions we're examining, and it's upon this stage that the drama between tangent vectors and functions unfolds: the result being that constant functions are immutable under the gaze of tangent vectors, bringing forth the zero value as explained previously.