Then understand the relationship between partial derivatives, directional derivatives, and gradients
My related notes:
1. Partial Derivative
2. Directional derivatives and gradient vectors
Personal summary:
1. The partial derivative reflects the rate of change of the multivariate function along the coordinate axis
2. The directional derivative reflects the rate of change of the multivariate function along any specified direction
3. The essence of the gradient is a vector, and the gradient direction is the maximum directional derivative (rate of change) of the function at a certain point In the direction of . If the direction is along the x-axis or y-axis, the obtained directional derivative is the partial derivative
1. Partial derivative (along the direction of the coordinate axis)
2. Directional derivative (along any direction)
3. Gradient (the direction with the largest directional derivative)
4. Two-dimensional gradient direction and three-dimensional gradient direction
The following content is quoted from: Intuitively understanding the relationship between “gradient” and “normal vector” – Zhihu @tetradecane
The gradient of a binary function is the normal vector of a two-dimensional contour.
The gradient of a ternary function is the normal vector of a three-dimensional isosurface.
5. The relationship between the three
Directional derivative = == Gradient⋅\cdot⋅Direction vector
(Remarks:The gradient is a linear combination of partial derivatives, i and j are the coordinate axis unit vectors)
Finally, use a picture to summarize:
