If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Multivariable calculus>Unit 2

Lesson 2: Gradient and directional derivatives

The gradient captures all the partial derivative information of a scalar-valued multivariable function. Created by Grant Sanderson.

## Want to join the conversation?

• What is a partial derivative operator?Especially what is operator?
• Operation is just a function. Just another name for it.
Operator is a symbol which correspondences to a function.
Example:
Addition is a operation. In other words, it is a function. It's domain is (R x R) (where R is a set of real numbers), and its' codomain is R. (you take two real numbers and obtain a result, one real number)
You can write it like this: +(5,3)=8. It's a familiar function notation, like f(x,y), but we have a symbol + instead of f. But there is other, slightly more popular way: 5+3=8. When there aren't any parenthesis around, one tends to call this + an operator. But it's all just words.

Partial derivative operator, nabla, upside-down triangle, is a symbol for taking the gradient, which was explained in the video.
Sidenote: (Sometimes the word "operator" is interchangeable with "operation", but you see this all the time. Words like "cook" (the person) and "(to) cook" are almost the same, because we tend to think of things that do the actions as the actions themselves)
• This dude is 3Blue1Brown isn't he?
• Yes! He is. His major was math at Stanford!
• What was the other name for gradient? At ish
• It is nabla or del. "del f" is how you would pronounce the gradient of f. (or grad f).
nabla is an upside-down Greek delta.
• He expressed the gradient like we do in matrix using the square brackets thingy. So gradient is a vector & a matrix?
• The gradient is only a vector. A vector in general is a matrix in the ℝˆn x 1th dimension (It has only one column, but n rows).
• For nabla, is the order of the components in the vector dependent on the order of the variables in the function call? For example, would:
``del f(y, x)``

be equal to:
``<df/dy, df/dx>``
?
Another example of my question would be:
``del f(x1, x2, x3, x4)``

translates to:
``<df/dx1, df/dx2, df/dx3, df/dx4>``
.
• Good question! At first it kind of seems an obvious thing to state, but we can't make assumptions in math now can we.

To answer your question, in my experience we always calculate the gradient in order of the operands, like you described.

Happy learning!
- Convenient Colleague
• At , when we take the derivative of f in respect to x, therefore take y = sin(y) as a constant, why doesn't it disappear in the derivative?
• A constant disappears in the derivative when it is added.
f(x) = c * g(x)
=> f'(x) = c *g'(x)
In this case g(x) = x² and c = sin(y)

f(x) = g(x) +c
Then we would have: f'(x) = g'(x)
When the constant is added, it disappears :)
• The function f (x,y) =x^2 * sin (y) is a three dimensional function with two inputs and one output and the gradient of f is a two dimensional vector valued function. So isn't he incorrect when he says that the dimensions of the gradient are the same as the dimensions of the function. I think it is always one less.
• The dimension of the gradient is always the same as the dimension of the input space.
This is due to the way we construct the gradient : we add a component for each variables.
In this case we have a two-dimensional input space, and therefore a two-dimensional gradient :)
• a minute ago what is Gradient??