Question

If \(\phi: \mathbb{R}^{k} \rightarrow \mathbb{R}\) is a \(C^{1}\) function, the graph of \(\phi\) is a \(k\)-dimensional \(C^{1}\) submanifold of \(\mathbb{R}^{k+1}\) parametrized by \(f(x)=(x, \phi(x))\). If \(A \subset \mathbb{R}^{k}\), the \(k\)-dimensional volume of the portion of the graph lying above \(A\) is $$ \int_{A} \sqrt{1+|\nabla \phi(x)|^{2}} d x $$ (First do the linear case, \(\phi(x)=a \cdot x\), Show that if \(T: \mathbb{R}^{k} \rightarrow \mathbb{R}^{k+1}\) is given by \(T x=(x, a \cdot x)\), then \(T^{*} T=I+S\) where \(S x=(a \cdot x) a\), and hence \(\operatorname{det}\left(T^{*} T\right)=\) \(1+|a|^{2}\). Hint: The determinant of a matrix is the product of its eigenvalues; what are the eigenvalues of \(S ?\) )

Short answer

\( \operatorname{det}(T^{*} T) = 1 + |a|^{2} \).

Step-by-step solution

Understand the Problem

We need to show that for a linear map \( T: \mathbb{R}^{k} \rightarrow \mathbb{R}^{k+1} \) defined by \( T x = (x, a \cdot x) \), the product \( T^{*} T \) is equal to \( I+S \), where \( Sx = (a \cdot x) a \), and then find the determinant of this matrix.

Define Matrix T

The map \( Tx = (x, a \cdot x) \) can be represented as a matrix transformation where the last row is \( a \). Thus, \( T \) is a matrix with an additional column added to an identity matrix representing the dot product with \( a \).

Compute \( T^{*} \)

The adjoint (or transpose) \( T^{*} \) for the transformation \( T: \mathbb{R}^{k} \rightarrow \mathbb{R}^{k+1} \) is obtained by transposing the matrix \( T \). This results in a matrix where the final row is replaced by the vector \( a \).

Compute \( T^{*} T \)

We calculate the product \( T^{*} T \). Since \( T^{*} \) has dimensions \((k+1) \times k \) and \( T \) has dimensions \( k \times (k+1) \), the product yields a matrix of dimensions \( k \times k \). The product yields an identity matrix (since \( x \) is mapped to itself) plus the projection matrix \( Sx = (a \cdot x)a \).

Verify \( T^{*} T = I + S \)

Verify the entries to confirm that \( T^{*} T = I + S \). The elements responsible for the projection are precisely derived from the dot product terms involving \( a \). This matches the form \((I + S)x = x + (a \cdot x)a\).

Calculate Eigenvalues of S

Since \( Sx = (a \cdot x)a \) which means it projects onto \( a \), its eigenvalues are \( |a|^{2} \) (due to projection magnitude along \( a \)) and 0 for the remaining dimensions perpendicular to \( a \).

Determine Determinant of \( T^{*} T \)

The determinant of \( T^{*} T = I + S \) is obtained by multiplying its eigenvalues. Here, one of the eigenvalues is \( 1+|a|^{2} \) (as \( S \) is rank 1), and all other eigenvalues are 1.

Key concepts

Submanifolds

In the world of differential geometry, a 'submanifold' refers to a space that is a lower-dimensional surface within a higher-dimensional manifold. Imagine a sheet of paper (2D) sitting in our 3D space. The sheet is a simple example of a submanifold.
Now, when we say that the graph of a function \(abla \phi(x)\) is a submanifold, we are considering the graph as a lower-dimensional "slice" in a larger dimensional space. Specifically, if we have a function \(abla \phi: \mathbb{R}^{k} \rightarrow \mathbb{R}^{k+1}\), its graph in \mathbb{R}^{k+1}\ is a \(k\)-dimensional surface. This is analogous to considering the curve drawn on a piece of paper.
Submanifolds are essential in understanding the properties of functions and how they behave when mapped into higher dimensions.

Volume of Graphs

The 'volume of graphs' concept is about measuring the "space" a surface created by a function occupies. Consider a graph of a function \(abla \phi(x)\), where \(A\) is the domain of interest in \mathbb{R}^{k}\. The volume of the graph above \(A\) is computed using the formula:
\[ \int_{A} \sqrt{1+|abla \phi(x)|^{2}} dx \]
Here's what's happening: \(abla \phi(x)\) represents the gradient or slope of the function \(\phi\). By incorporating \(|abla \phi(x)|^{2}\), we effectively adjust for how much the graph "leans" or "steeps" over \(A\). This integral essentially "measures" the surface's area, factoring in these slopes. This concept is vital when studying how functions behave in terms of space and understanding phenomena such as surface tension or energy states of a physical system.

Matrix Determinants

A determinant is a fundamental concept in linear algebra, used for understanding matrix properties, including transformations they represent. For our function \(abla \phi(x)\), when we represent it as a matrix transformation \(\boldsymbol{T}\), the determinant provides insight into the volume transformation upon applying this matrix.
In particular, for the exercise, we explore \(\boldsymbol{T}: \mathbb{R}^{k} \rightarrow \mathbb{R}^{k+1}\) given by \(\boldsymbol{T}x = (x, a \cdot x)\). We find that:
\[ T^{*}T = I + S \]
Where \(\boldsymbol{S}x = (a \cdot x)a\). The determinant of \(T^{*}T\) is \(\operatorname{det}(T^{*}T) = 1+|a|^{2}\). Intuitively, this tells us about the "size change" induced by the transformation \(T\), particularly the expansion or compression along the vector \(a\). Calculating determinants is crucial in various applications, such as robotics for determining volume changes or control systems in stability analysis.

Gradient and Eigenvalues

Gradients and eigenvalues are pivotal in understanding how functions change and are transformed. The gradient \(abla \phi(x)\) is a vector of partial derivatives, representing the slope or rate of change of the function \(\phi\) at any given point. It is crucial for finding the steepest descent or ascent, often used in optimization problems.
In the context of linear transformations like our matrix \(\boldsymbol{T}\), eigenvalues offer insight into how the transformation stretches or compresses space. In the exercise, it's evidenced through \(\boldsymbol{S}x = (a \cdot x)a\).
The eigenvalues of \(\boldsymbol{S}\) were identified as \(|a|^{2}\) and zeros for other dimensions. These eigenvalues reveal how space is stretched specifically along the direction of \(a\) and remain unchanged perpendicular to it.
Understanding these components helps in predicting the behavior of complex systems, whether in physics, economics, or any field involving multi-variable functions.