Is a gradient just jacobian with a single row?

Math queston

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Yes, you're on the right track. In vector calculus, the gradient is a key operation which takes a scalar-valued function and creates a vector-valued function (vector field). In essence, the gradient of a function gives you a vector pointing in the direction of the greatest rate of increase of the function, and the magnitude of that vector is the rate of increase in that direction. On the other hand, the Jacobian matrix is a matrix of all the first-order partial derivatives of a vector-valued function, a generalization of gradient. In simpler terms, the Jacobian matrix represents how changes in the input of a function result in changes in its output. If your function has only one component and is real-valued, the Jacobian for your function reduces down to a single row; this row is what we generally refer to as the gradient. But remember, for functions with multiple outputs, the Jacobian will be a whole matrix, not just a single row. It encapsulates information about the rate of change in every direction, for every output direction. Thus, in a sense, you can say a gradient is a Jacobian with a single row for a function from ℝᵈ to ℝ. This connection between the gradient and the Jacobian is a good starting point to understanding how higher dimensions work, and how we can generalize concepts from 3 dimensions to any number of dimensions. If you're interested in the mathematics of machine learning, this concept is significant because machine learning often works with large dimensions.
Answered on August 3, 2023.
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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.
Answered on August 24, 2023.
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