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I don't really get what's the difference between them. What does each thing physically and mathematically signify? Aren't both things just a dot product with the del operator?

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    $\begingroup$ Taking the gradient of a scalar field produces a vector field, while divergence applied to a vector field produces a scalar field $\endgroup$
    – DanDan面
    Mar 31 at 4:55
  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community
    Mar 31 at 4:59
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    $\begingroup$ Aren't both things just a dot product with the del operator? No. The gradient doesn’t involve any dot product. It can’t because it only involves one “vectorial” thing — namely $\nabla$ — not two. $\endgroup$
    – Ghoster
    Mar 31 at 5:03

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Mathematically, the gradient is a property of a scalar function $f:\mathbb R^n\to\mathbb R$, found as

$$\mathrm{grad} (f)=\nabla f=\begin{bmatrix}\frac{\partial f}{\partial x^1}\\\frac{\partial f}{\partial x^2}\\ \frac{\partial f}{\partial x^3}\\\vdots\end{bmatrix}.$$

In physical terms you can think of it as the equivalent of the derivative of a function of one variable. It is "the derivative" or "the slope" in higher dimensions, so to speak. For instance, for a function of two variables $f:\mathbb R^2\to\mathbb R$, which represents a surface when plotted, the gradient is a vector arrow that always points in the steepest direction from any point.

The divergence on the other hand is a property of a vector function $V:\mathbb R^n\to\mathbb R^n$, which more specifically is called a vector field, and is found as

$$\mathrm{Div}(\mathbf V)=\nabla \cdot \mathbf V=\frac{\partial V^1}{\partial x^1}+\frac{\partial V^2}{\partial x^2}+\frac{\partial V^3}{\partial x^3}+\cdots$$

Physically, if you think of a vector field as representing e.g. the wind velocity at every point, then the divergence can be thought of as indicating a local expansion rate (do the points tend to move away from or closer to each other at a point - i.e., does the density decrease or increase at a point?).

Both the gradient concept and the divergence concept can be defined using the nabla operator $\nabla$, which might be what you are referring to. But note that this operator is used in simple scalar multiplication in the case of the gradient because we there are dealing with a scalar function, whereas it is used in a dot product in the case of the divergence where we are dealing with a vector product. Also note have the outputs are entirely different types of objects that have different dimensions - the gradient is a vector whereas the divergence is a scalar. So I would not try to think of them as related.

Physically, scalar functions and vector fields should better be thought of as entirely different mathematical concepts that in turn can represent entirely different and essentially unrelated physical concepts, such as mountain surfaces vs. wind directions.

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