- #1
member 428835
hey pf!
i have a few question about the physical intuition for divergence, gradient, and curl. before asking, i'll define these as i have seen them (an intuitive definition).
\text{Divergence} \:\: \nabla \cdot \vec{v} \equiv \lim_{V \to 0} \frac{1}{V} \oint_A \hat{n} \cdot \vec{v} da
\text{Curl} \:\: \nabla \times \vec{v} \equiv \lim_{V \to 0} \frac{1}{V} \oint_A \hat{n} \times \vec{v} da
\text{Gradient} \:\: \nabla \vec{v} \equiv \lim_{V \to 0} \frac{1}{V} \oint_A \hat{n} \vec{v} da
where ##V## is the volume, ##\vec{v}## is a vector field and the rest (i think) is evident.
as for the divergence, i understand that ##\hat{n} \cdot \vec{v} da## is a volumetric flow rate through a differential surface element. thus the integral is the total volumetric flow rate, and dividing by volume and the limit gives us a nice understanding that divergence is expansion/contraction of a vector field.
my understanding is predicated on the understanding of the dot product, namely ##\hat{n} \cdot \vec{v} da##. however, the curl uses a cross product. i understand a cross product to be a vector orthogonal to two given vectors. thus, ##\hat{n} \times \vec{v} da## seems to be some new vector in a field such that it is tangent to the surface at any point. if we add all these vectors ##\hat{n} \times \vec{v} da## up we should get some kind of body rotation about a point, which i think parallels the general understanding of curl.
however, what on Earth do we do about the dyadic product ##\hat{n} \vec{v}## embedded in the definition of gradient? i really don't have a physical interpretation of what is happening here, and thus i really don't physically understand ##\nabla \vec{v}##.
please help!
i have a few question about the physical intuition for divergence, gradient, and curl. before asking, i'll define these as i have seen them (an intuitive definition).
\text{Divergence} \:\: \nabla \cdot \vec{v} \equiv \lim_{V \to 0} \frac{1}{V} \oint_A \hat{n} \cdot \vec{v} da
\text{Curl} \:\: \nabla \times \vec{v} \equiv \lim_{V \to 0} \frac{1}{V} \oint_A \hat{n} \times \vec{v} da
\text{Gradient} \:\: \nabla \vec{v} \equiv \lim_{V \to 0} \frac{1}{V} \oint_A \hat{n} \vec{v} da
where ##V## is the volume, ##\vec{v}## is a vector field and the rest (i think) is evident.
as for the divergence, i understand that ##\hat{n} \cdot \vec{v} da## is a volumetric flow rate through a differential surface element. thus the integral is the total volumetric flow rate, and dividing by volume and the limit gives us a nice understanding that divergence is expansion/contraction of a vector field.
my understanding is predicated on the understanding of the dot product, namely ##\hat{n} \cdot \vec{v} da##. however, the curl uses a cross product. i understand a cross product to be a vector orthogonal to two given vectors. thus, ##\hat{n} \times \vec{v} da## seems to be some new vector in a field such that it is tangent to the surface at any point. if we add all these vectors ##\hat{n} \times \vec{v} da## up we should get some kind of body rotation about a point, which i think parallels the general understanding of curl.
however, what on Earth do we do about the dyadic product ##\hat{n} \vec{v}## embedded in the definition of gradient? i really don't have a physical interpretation of what is happening here, and thus i really don't physically understand ##\nabla \vec{v}##.
please help!