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Finding logarithmic gradient in the hyperbolic gradient
I’m trying to find the logarithmic gradient of the function $f(x,y,z)=x+y+z$ using the relation between the gradient and the exponential map
We have that $\frac{\partial f}{\partial x}=1\Longrightarrow \exp_p(v)=p+v$ by the chain rule and $\frac{\partial f}{\partial v}=\frac{\partial f}{\partial t}=3e^t$ so $\exp_p(v)\exp_p(-t)=p+v+3te^t$
Now I’m not sure how to proceed, I know that $
abla f=\exp_p(v)\exp_p(-t)
abla g$ but if I write $g(t)=e^t$ then the gradient becomes $g(v+t)$ which is not the logarithmic gradient I need. Am I thinking about this in