Can we obtain the equation of continuity in the form as in the attached image, by using the local and partial rates of change formula for vectors and scalars(please refer the other image)? If so, can I have a detailed explanation on how to start with the formula of local and partial rates of change to obtain, the equation of continuity as in the form given in the relevant image? Thank you!

Can we obtain the equation of continuity in the form as in the attached image, by using the local and partial rates of change formula for vectors and scalars(please refer the other image)? If so, can I have a detailed explanation on how to start with the formula of local and partial rates of change to obtain, the equation of continuity as in the form given in the relevant image? Thank you!

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Can we obtain the equation of continuity in the form as in the attached image, by using the local and partial rates of change formula for vectors and scalars(please refer the other image)?

If so, can I have a detailed explanation on how to start with the formula of local and partial rates of change to obtain, the equation of continuity as in the form given in the relevant image?

Thank you!