Let $\varphi:(M,g,\nabla)\to\mathbb{R}^n$ be a smooth embedding of a convex hypersurface. I want to explicitly calculate $$\langle \varphi,\varphi_{\ast}(\nabla\|\varphi\|^2)\rangle.$$ In particular, I want to know if $\langle \varphi,\varphi_{\ast}(\nabla\|\varphi\|^2)\rangle\geq0.$ Here $\langle,\rangle$ denotes the Euclidean inner product.
Remark: A version of this question was posted by me earlier, however, since it was misrepresented, I decided to erase.
I assume that ($g, \nabla$) are induced metric and connection from the embedding $\varphi$. Note that $\nabla \| \varphi\|^2 = 2 \varphi ^\top$, the tangential part of $\varphi$. Thus
$$\langle \varphi , \nabla \|\varphi\|^2 \rangle = 2\|\varphi^\top\|^2 \geq 0.$$
(It's true for any immersion $\varphi$)
Remark: To show that $\nabla \| \varphi\|^2 = 2 \varphi ^\top$, let $x\in M$ and $e_1, \cdots e_{n-1}$ be an orthonormal bases of $T_xM$. Then in general for any smooth function $f:M \to \mathbb R$,
$$\nabla f = \sum_{i=1}^{n-1} (\nabla_{e_i} f) \ e_i.$$
Hence
$$\nabla \|\varphi\|^2 = \sum (\nabla_{e_i} \|\varphi\|^2) \ e_i = 2 \sum \langle \varphi, e_i\rangle \ e_i = 2\varphi^\top. $$