A vector of partial derivatives pointing in the direction of steepest increase.
The gradient ∇f=(f_x,f_y,f_z) points in the direction of steepest increase and its magnitude is the maximum rate. It is perpendicular to level curves. Heat flows opposite to ∇T and gradient descent optimises functions.
∇(x²+y²) at (1,2) = (2,4).
The gradient tells you how to change inputs to increase an output fastest, which drives optimisation and learning algorithms.
A vector of partial derivatives pointing in the direction of steepest increase.
The gradient tells you how to change inputs to increase an output fastest, which drives optimisation and learning algorithms.
It points in the direction of steepest increase, and its magnitude gives the rate of that increase.
Along a level curve the function does not change, so the gradient, which measures change, must be perpendicular to it.
Typical applications include finding the steepest descent direction when tuning design parameters, computing the direction of a heat flux from a temperature field and calculating the surface normal of a level set in a CAD model.
Study "Partial derivatives" first (topic 41 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Directional derivative = ∇f·u. Normal vector to a surface. Force = -∇(potential energy).
∇(x²+y²) at (1,2) = (2,4).
The next topic is "Multiple integrals", which builds directly on this one.
It is classed as Intermediate in Engineering Mathematics (topic 42 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.
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