**Update:** It is almost sure that the expression of $\kappa$ in coordinates given in the book, namely

$$\kappa(u_t)=h_{\alpha\beta}\frac{\partial u_t^\alpha}{\partial x^i}\frac{\partial u_t^\beta}{\partial x^j}$$

is a typo. I have edited the question and given my version of this below.

In the book *Variational Problems in Geometry* by Seiki Nishikawa, two Weitzenbock formulae are given (Proposition 4.2, page 124):

Let $u\in C^\infty(M\times(-\delta,\delta),N)$, where $M,N$ are Riemannian manifolds, be a solution to the equation $\frac{\partial u}{\partial t}=\tau(u_t)$, where $\tau(u_t)$ is the tension field of $u_t(-):=u(-,t):M\to N$. Then we have

$$\frac{\partial e(u_t)}{\partial t}=\Delta e(u_t)-|\nabla\nabla u_t|^2-\sum_{i=1}^m\left\langle du_t\left(\sum_{j=1}^mRic^M(e_i,e_j)e_j\right),du_t(e_i)\right\rangle\\ +\sum_{i,j=1}^mR^N(du_t(e_i),du_t(e_j),du_t(e_j),du_t(e_i)$$

$$\frac{\partial\kappa(u_t)}{\partial t}=\Delta\kappa(u_t)-\left|\nabla\frac{\partial u_t}{\partial t}\right|^2+\sum_{i=1}^mR^N\left(du_t(e_i),\frac{\partial u_t}{\partial t},\frac{\partial u_t}{\partial t},du_t(e_i)\right)$$

where $Ric^M$ is the Ricci tensor on $M$, $R^N$ the curvature tensor on $N$, $\Delta$ the Laplace operator on $M$, $e_i$ an orthonormal basis of tangent spaces of $M$, and $\left\langle,\right\rangle$ is the inner product on the tangent spaces of $N$.

The quatities $e(u_t)$, $\kappa(u_t)$ are defined on page 123 of the book

$$e(u_t)=\frac{1}{2}|du_t|^2=\frac{1}{2}g^{ij}h_{\alpha\beta}\frac{\partial u_t^\alpha}{\partial x^i}\frac{\partial u_t^\beta}{\partial x^j}$$

$$\kappa(u_t)=\frac{1}{2}\left|\frac{\partial u_t}{\partial t}\right|^2=\frac{1}{2}g^{ij}u_{t;ij}^\alpha g^{rs}u_{t;rs}^\beta h_{\alpha\beta}$$

where $g_{ij}$, $h_{\alpha\beta}$ are the components of the metric tensor on $M$, $N$ respectively, and a semicolon together with subscripts denotes covariant derivatives. **Note that the coordinate representation of $\kappa(u_t)$ is not given as a definition, but a result of calculation under the assumption that
$$\frac{\partial u_t}{\partial t}=\tau(u_t)=g^{ij}u_{t;ij}^\alpha\frac{\partial}{\partial y\alpha}$$ holds**.

**Question:**

The formula about $e(u_t)$ has a complete proof in the book. The book says the second one can be proven in a similar manner to the first one. However, the original coordinate representation of $\kappa(u_t)$ given in the book must be wrong (see the top of this post), since the RHS has indexes $i,j$ while the LHS does not. I have given my version of $\kappa(u_t)$ in coordinates in the question, which is nothing similar to the expression of $e(u_t)$ in coordinates. And I now really don't know how to prove the second Weitzenbock formula "in a similar manner to the first one". Can you provide a sketch of proof of the second formula? A reference is also okay.