# Partial derivatives of bilinear Sampling

I'm going over the paper Spatial Transformer Networks (link). On section 3.3 they introduce bilinear sampling kernel (Eq. 5)
$$V_{i}^{c}=\sum_{n}^{H} \sum_{m}^{W} U_{n m}^{c} \max \left(0,1-\left|x_{i}^{s}-m\right|\right) \max \left(0,1-\left|y_{i}^{s}-n\right|\right),$$ and it's partial derivative (Eq. 6)
$$\frac{\partial V_{i}^{c}}{\partial U_{n m}^{c}}=\sum_{n}^{H} \sum_{m}^{W} \max \left(0,1-\left|x_{i}^{s}-m\right|\right) \max \left(0,1-\left|y_{i}^{s}-n\right|\right).$$

Since we take the derivative w.r.t. $$U^c_{nm}$$, it seems to me like the partial derivative should be $$\frac{\partial V_{i}^{c}}{\partial U_{n m}^{c}}=\max \left(0,1-\left|x_{i}^{s}-m\right|\right) \max \left(0,1-\left|y_{i}^{s}-n\right|\right),$$ otherwise, the indices $$m,n$$ of the lhs is meaningless, and all of the partial derivatives will be identical.

Did I miss anything regarding the definition of $$V^c_i$$, $$x^s_i$$, or $$U^c_{nm}$$?