RE: Is a gradient just jacobian with a single row?
No, a gradient isn't exactly the Jacobian with a single row. Both are related but are slightly different entities in vector calculus. The gradient is a vector operation that operates on a scalar function to produce a vector whose magnitude is the maximum rate of change of the function. More intuitively, the gradient of a function points in the direction of the greatest increase of the function and its magnitude is the slope of the function in that direction. It is usually represented in the Cartesian coordinates as (∂f/∂x, ∂f/∂y, ∂f/∂z) for a three-dimensional function. The Jacobian, on the other hand, is a matrix that represents all the first order partial derivative of a vector-valued function. In case of a function with multiple variables, it would consist multiple rows and columns. When the function being considered is scalar (i.e., it has only one component), the Jacobian matrix reduces to a single row, and it aligns with the definition of the gradient. In such case, you can imagine the gradient as the Jacobian of a scalar field. In conclusion, for scalar functions, the gradient can be seen as a special case of the Jacobian, having only one row indeed. But in a broader sense, they are two distinct mathematical concepts serving different purposes. I hope this insight serves not only to answer your immediate query, but also helps future web surfers looking to distinguish these two concepts.