Summary table of multivariable derivatives
This page is a summary table of multivariable derivatives.
- TODO maybe good to have separate rows for evaluated and pre-evaluated versions, for things that are functions/can be applied
Single-variable real function
For comparison and completeness, we give a summary table of the single-variable derivative. Let be a single-variable real function.
|Derivative of at||or or||In the most general multivariable case, will become a linear transformation, so analogously we may wish to talk about the single-variable as the function defined by , where on the left side "" is a function and on the right side "" is a number. If "" is a function, we can evaluate it at to recover the number: . This is pretty confusing, and in practice everyone thinks of "" in the single-variable case as a number, making the notation divergent; see Notational confusion of multivariable derivatives § The derivative as a linear transformation in the several variable case and a number in the single-variable case for more information.|
Real-valued function of Rn
Let be a real-valued function of .
|Partial derivative of with respect to its th variable||or or or or||Here is the th vector of the standard basis, i.e. the vector with all zeroes except a one in the th spot. Therefore can also be written when broken down into components.|
|Gradient at||or||or the vector such that|
|Directional derivative in the direction of||or||When , this reduces to the th partial derivative.|
I think in this case, since coincides with , people don't usually define the derivative separately. For example, Folland in Advanced Calculus defines differentiability but not the derivative! He just says that the vector that makes a function differentiable is the gradient.
TODO: answer questions like "Is the gradient the derivative?"
Vector-valued function of R
Let be a vector-valued function of . A parametric curve (or parametrized curve) is an example of this. Since the function is vector-valued, some authors use a boldface letter like .
|Velocity vector at||or|
Note the absence for partial/directional derivatives. There is only one variable with respect to which we can differentiate, so there is no direction to choose from.
Vector-valued function of Rn
Let be a vector-valued function of . Since the function is vector-valued, some authors use a boldface letter like .
|Partial derivative with respect to the th variable||or or or or|
|Directional derivative in the direction of||or|
|Total or Fréchet derivative (sometimes just called the derivative) at point||or or||The linear transformation such that||The derivative at a given point is a linear transformation. One might wonder then what the derivative (without giving a point) is, i.e. what meaning to assign to "" as we can in the single-variable case. Its type would have to be or more specifically . Also the notation is slightly confusing: if the total derivative is a function, what happens if ? We see that , so the single-variable derivative isn't actually a number! To get the actual slope of the tangent line, we must evaluate the function at : . Some authors avoid this by using different notation in the general multivariable case. Others accept this type error and ignore it.|
|Derivative matrix, differential matrix, Jacobian matrix at point||or||Since the total derivative is a linear transformation, and since linear transformations from to have a one-to-one correspondence with real-valued by matrices, the behavior of the total derivative can be summarized in a matrix; that summary is the derivative matrix. Some authors say that the total derivative is the matrix. TODO: talk about gradient vectors as rows.|
Note the absence of the gradient in the above table. The generalization of the gradient to the case is the derivative matrix.
- Notational confusion of multivariable derivatives
- calculus:Relation between gradient vector and partial derivatives
- calculus:Relation between gradient vector and directional derivatives
- calculus:Directional derivative
- Summary table of probability terms
- Tao, Terence. Analysis II. 2nd ed. Hindustan Book Agency. 2009.
- Folland, Gerald B. Advanced Calculus. Pearson. 2002.
- Pugh, Charles Chapman. Real Mathematical Analysis. Springer. 2010.