Here ,the Jacobian Matrix ,takes 3 functions that are made of r, θ, Φ and helps it to transform to x ,y ,z (ie) it basically acts like a transformation matrix ,my question is
1)Is my assumption of changing from Spherical to Cartesian coordinates correct ,Is that the function of Jacobian Matrix here?
2)If it is ,then shouldn't the LHS to the equal sign have x,y,z terms , kind of like basis vectors ,because it is taking in r,θ,Φ and transforming to x,y,z.
Take the function $f:U:=\mathbb R^+\times(0,\pi)\times(0,2\pi)\to \mathbb R^3$ such that $$f(r,\theta,\phi)=(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta)=(x(r,\theta,\phi),y(r,\theta,\phi),z(r,\theta,\phi)).$$ This map induces a linear function on tangent spaces that acts on tangent vectors: $$f_{*p}:T_pU\to T_{f(p)}\mathbb R^3.$$ If you now take the basis of the tangent space $T_pU$, which is $\{\partial_{r},\partial_{\theta},\partial_{\phi}\}$, we can find their images in terms of $x,y,z$ coordinates through the computation of $f_{*p}(\partial_r),f_{*p}(\partial_{\theta}),f_{*p}(\partial_{\phi})$.
For example, $$f_{*p}(\partial _r)=\dfrac{\partial x(r,\theta,\phi)}{\partial r}\vert_p\cdot\dfrac{\partial}{\partial x}\vert_{f(p)}+\dfrac{\partial y(r,\theta,\phi)}{\partial r}\vert_p\cdot\dfrac{\partial}{\partial y}\vert_{f(p)}+\dfrac{\partial z(r,\theta,\phi)}{\partial r}\vert_p\cdot\dfrac{\partial}{\partial z}\vert_{f(p)}$$ and do the same for the other two coordinates $\theta$ and $\phi$.
In general, the expression for $f_{*}(\partial_r)$ is given by $$f_*(\partial_r)=\dfrac{\partial x}{\partial r}\circ f^{-1}\cdot\dfrac{\partial}{\partial x} +\dfrac{\partial y}{\partial r}\circ f^{-1}\cdot\dfrac{\partial}{\partial y}+\dfrac{\partial z}{\partial r}\circ f^{-1}\cdot\dfrac{\partial}{\partial z}$$ because in this way we'll have the push foward of tangent vectors expressed in $x,y,z$ coordinates ($f$ has to be a diffeomorphism). The Jacobian of $f(r,\theta,\phi)$ evaluated at a point $p\in U$ acts on tangent vectors $T_pU\ni v=a\partial_r+b\partial_{\theta}+c\partial_{\phi}$ as the matrix of representation of the linear operator $f_{*p}$.