This may be borderline for this site, but here goes: What you are looking for may be the notion of weak derivative. A function $f$ defined on some $d$-dimensional set $\Omega$ has a weak partial derivative with respect to the $i$-th coordinate if there exists a function $g_i$ if for all smooth functions $\phi$ with compact support in $\Omega$ it holds that $$ \int_\Omega f(x)\partial_i \phi(x) dx = -\int_\Omega g_i(x)\phi(x)dx $$ So this looks very much like the distributional derivative and in fact one can say that $f$ has weak partial derivatives if the respective distributional derivatives are can be identified with locally integrable functions.

Similarly, you can define other differentiable operators: A locally integrable (vector-valued) function $\vec g$ is a weak gradient of $f$ if for all vector valued smooth function $\vec \phi$ with compact support it holds that $$ \int_\Omega f(x)\operatorname{div}\vec\phi(x) dx = -\int_\Omega \vec g(x)\cdot\vec\phi(x)dx $$ and so on.

However, this is still not satisfying, since you need a certain notion of integral over the boundary of merely integrable functions. The theory you are looking for is the theory of Sobolev spaces and their traces. For the boundary integrals you may even want to consider integrals with respect to the Hausdorff measure. One reference is the book "Geometric integration theory" by Krantz and Parks.

This may be borderline for this site, but here goes: What you are looking for may be the notion of weak derivative. A function $f$ defined on some $d$-dimensional set $\Omega$ has a weak partial derivative with respect to the $i$-th coordinate if there exists a function $g_i$ such that for all smooth functions $\phi$ with compact support in $\Omega$ it holds that $$ \int_\Omega f(x)\partial_i \phi(x) dx = -\int_\Omega g_i(x)\phi(x)dx. $$ This looks very much like the distributional derivative and in fact one can say that $f$ has weak partial derivatives if the respective distributional derivatives can be identified with locally integrable functions.

Similarly, you can define other differential operators: A locally integrable (vector-valued) function $\vec g$ is a weak gradient of $f$ if for all vector valued smooth functions $\vec \phi$ with compact support it holds that $$ \int_\Omega f(x)\operatorname{div}\vec\phi(x) dx = -\int_\Omega \vec g(x)\cdot\vec\phi(x)dx $$ and so on.

However, this is still not satisfying, since you need a certain notion of integral over the boundary of merely integrable functions. The theory you are looking for is the theory of Sobolev spaces and their traces. For the boundary integrals you may even want to consider integrals with respect to the Hausdorff measure. One reference is the book "Geometric integration theory" by Krantz and Parks.

This may be borderline for this site, but here goes: What you are looking for may be the notion of weak derivative. A function $f$ defined on some $d$-dimensional set $\Omega$ has a weak partial derivative with respect to the $i$-th coordinate if there exists a function $g_i$ if for all smooth functions $\phi$ with compact support in $\Omega$ it holds that $$ \int_\Omega f(x)\partial_i \phi(x) dx = -\int_\Omega g_i(x)\phi(x)dx $$ So this looks very much like the distributional derivative and in fact one can say that $f$ has weak partial derivatives if the respective distributional derivatives are can be identified with locally integrable functions.

Similarly, you can define other differentiable operators: A locally integrable (vector-valued) function $\vec g$ is a weak gradient of $f$ if for all vector valued smooth function $\vec \phi$ with compact support it holds that $$ \int_\Omega f(x)\operatorname{div}\vec\phi(x) dx = -\int_\Omega \vec g(x)\cdot\vec\phi(x)dx $$ and so on.

However, this is still not satisfying, since you need a certain notion of integral over the boundary of merely integrable functions. The theory you are looking for if the theory of Sobolev spaces and their traces. For the boundary integrals you may even want to consider integrals with respect to the Hausdorff measure. One reference is the book "Geometric integration theory" by Krantz and Parks.

This may be borderline for this site, but here goes: What you are looking for may be the notion of weak derivative. A function $f$ defined on some $d$-dimensional set $\Omega$ has a weak partial derivative with respect to the $i$-th coordinate if there exists a function $g_i$ if for all smooth functions $\phi$ with compact support in $\Omega$ it holds that $$ \int_\Omega f(x)\partial_i \phi(x) dx = -\int_\Omega g_i(x)\phi(x)dx $$ So this looks very much like the distributional derivative and in fact one can say that $f$ has weak partial derivatives if the respective distributional derivatives are can be identified with locally integrable functions.

Similarly, you can define other differentiable operators: A locally integrable (vector-valued) function $\vec g$ is a weak gradient of $f$ if for all vector valued smooth function $\vec \phi$ with compact support it holds that $$ \int_\Omega f(x)\operatorname{div}\vec\phi(x) dx = -\int_\Omega \vec g(x)\cdot\vec\phi(x)dx $$ and so on.

However, this is still not satisfying, since you need a certain notion of integral over the boundary of merely integrable functions. The theory you are looking for is the theory of Sobolev spaces and their traces. For the boundary integrals you may even want to consider integrals with respect to the Hausdorff measure. One reference is the book "Geometric integration theory" by Krantz and Parks.