# Jacobian of a skalar function with multi-dimentional vector input

I am trying to compute the Jacobian of $$f : \mathbb{R}^{8} \rightarrow \mathbb{R}$$, where:

$$f(\vec{x})= g(T(\vec{x}))= g(\vec{\mathbf{c}})=\Biggl| \|\mathbf{V}\|_{2}^{2} - \|\mathbf{A} \cdot c\|_{2}^{2} \Biggr| + \| \mathbf{c}_{r e f}-\mathbf{c} \|_{2} + \Biggl | 2-\|\mathbf{c}\|_{2} \Biggr | = g_{1}+g_{2}+g_{3}$$

and:

$$\vec{\mathbf{c}}=T(\vec{x}) = \left[x_{1}.exp(i.x_{5}), x_{2}.exp(i.x_{6}), x_{3}.exp(i.x_{7}), x_{4}.exp(i.x_{8}) \right] ^ { T}$$

After some research I ended up with the following:

$$\nabla g_{1}(c) = \frac{1}{\|Ac\|} c^T A^T A$$ based on this

$$\nabla g_{2}(c) = {1 \over 2\|c_{ref}-c\|_2}(2 (c_{ref}-c)^T) I = {c_{ref} -c \over \|c_{ref}-c\|_2}$$ based on this

$$\nabla g_{3}(c) = {-1.{c \over \|c\|_2}}$$ based on this

$$\nabla g(c) = \nabla g_{1}(c) + \nabla g_{2}(c) + \nabla g_{3}(c)$$

So my questions are the following:

• Is my approach correct?
• How to continue and go about the absolute value operators?

I am very new to this so any hints and advises are much appreciated. Thank you in advance.