$Vector-valued functions and Jacobian matrix$

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$Vector-valued functions and Jacobian matrix$

Since multivariable vector-valued functions are not covered in Stewart's Calculus I would like to clarify if I understand them correctly. So multivariable vector-valued function looks like this, right? $$f(x,y) = v<f1(x,y),f2(x,y)>$$
And Jacobian matrix would look like:
$$\begin{bmatrix} f1_x(x,y) & f1_y(x,y) \\ f2_x(x,y) & f2_y(x,y) \end{bmatrix} $$
Right?

Calculus

Babaduras

106

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Yes, a multivariable vector-valued function takes multiple input variables and produces a vector of outputs. It is typically represented as follows:

$\mathbf{f}(x, y) = \begin{bmatrix} f_1(x, y) \\ f_2(x, y) \end{bmatrix}$

In this notation, each component of the vector is a separate scalar-valued function. The Jacobian matrix for this vector-valued function is indeed a 2x2 matrix, and it represents the partial derivatives of each component with respect to the input variables:

$\mathbf{J}(\mathbf{f})(x, y) = \begin{bmatrix} \frac{\partial f_1}{\partial x}(x, y) & \frac{\partial f_1}{\partial y}(x, y) \\ \frac{\partial f_2}{\partial x}(x, y) & \frac{\partial f_2}{\partial y}(x, y) \end{bmatrix}$

Kav10

1.9K

Babaduras

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I don’t see any difference between Jacobian matrix you wrote and mine, except in way of writing partial derivative, is my matrix wrong?
- Kav10
  
  +1
  
  No, yours is correct. I write it like the partial derivative for better understaing. You got all those correct.
- Babaduras
  
  +1
  
  Ok, thank you.

$Vector-valued functions and Jacobian matrix$

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