Gradient

In Euclidean coordinates, partial derivatives form a vector describing directional change.

\nabla f=(\partial_1 f,\ldots,\partial_n f)

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Different ways to understand it

Gradient
Khan Academy · English · Explanation
Connection evidence
Reviewed English material v3: card "Gradient as packed partial-derivative information" at 0:41 — Before deriving any formula, the clip defines the gradient operationally: it is the vector that collects all the partial derivative information of a function into one object.
Change of variables, Jacobian determinant, Double and triple integrals｜701号时钟草
701号时钟草 · English · Explanation
Connection evidence
Reviewed English material v1: card "Jacobian Matrix and Gradient of a Multivariable Function" at 12:30 — When the codomain dimension of a multivariable function $u=f(x,y,z)$ is 1, its Jacobian matrix has only one row and is formally equivalent to the gradient vector $\nabla f$ of the function. Both can be regarded as the derivative of that multivariable function.
Lagrange Multipliers | Geometric Meaning & Full Example
Dr. Trefor Bazett · English · Explanation
Connection evidence
Reviewed English material v1: card "Gradient is normal to a level curve" at 5:51 — The clip recalls the standard fact that the gradient vector is normal to a level curve. In the example, arrows labeled $\nabla g$ are drawn perpendicular to the constraint circle at the tangency points, identifying the normal direction to $g(x,y)=0$ .
Method of Characteristics 1: Constant Coefficients
Partial Differential Equations · English · Explanation
Connection evidence
Reviewed English material v1: card "Gradient of a two-variable function" at 2:35 — The lecturer defines the gradient of u as the vector of its first partial derivatives. This definition is needed to understand the compact vector form of the PDE.
What Does the Gradient Vector Mean Intuitively?
The Math Sorcerer · English · Explanation
Connection evidence
Reviewed English material v1: card "Gradient definition for $f(x,y)$ " at 0:38 — The gradient is introduced as a vector made from the first-order partial derivatives of f with respect to x and y. On the board it is written as an ordered pair.
Dimensionality Reduction with PCA: How Principal Component Analysis Works
Infomity · English · Explanation
Connection evidence
Reviewed English material v2: card "Gradient of a Vector" at 6:03 — In matrix calculus, the gradient of a scalar function $f$ with respect to a $d$ -dimensional vector $\mathbf{x}$ is defined as a column vector containing all the partial derivatives of $f$ with respect to each component of $\mathbf{x}$ . The transpose symbol $T$ converts the row of partial derivatives into a column vector. This concept is fundamental for understanding optimization and advanced mathematical modeling.
Directional derivative
Khan Academy · English · Application
Connection evidence
Reviewed English material v1: card "Gradient Dot Product Form" at 5:25 — The directional derivative equals the dot product of the direction vector and the gradient vector, offering a compact geometric interpretation.

Questions and answers about Gradient

How do you expand the dot product (5,2)·∇u?

To expand (5,2)·∇u, substitute ∇ $u = (u_x, u_y)$ and apply the definition of the Euclidean dot product: multiply corresponding components and add them. This yields $5u_x + 2u_y$ .

∇ $u = (u_x, u_y)$ .
The dot product is defined as the sum of the products of corresponding components.

Example: The lecturer explains that the dot product multiplies the first components and adds the second components multiplied by each other. The board shows (5,2)·∇ $u=0$ and the spoken expansion gives $5u_x+2u_y$ .

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What does the gradient ∇u represent for a function $u(x,y)$ ?

The gradient ∇u is the vector whose components are the first partial derivatives of u. For a function of two variables, ∇ $u = (u_x, u_y)$ , where $u_x$ is the partial derivative with respect to x and $u_y$ is the partial derivative with respect to y.

u is a scalar function of two variables x and y.
u has partial derivatives with respect to x and y.

Example: The lecturer defines ∇u as the vector whose first component is the partial derivative of u with respect to x and whose second component is the partial derivative of u with respect to y. The board shows ∇ $u = (u_x, u_y)$ .

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Why can the directional derivative be written as the dot product of the direction vector and the gradient?

The directional derivative formula a*(∂f/∂x) + b*(∂f/∂y) structurally matches the definition of a dot product between the vector [a,b] and the vector [∂f/∂x, ∂f/∂y]. Since the second vector is defined as the gradient ∇f, the expression simplifies to w·∇f.

The function $f(x,y)$ is differentiable.
The direction vector is $w=[a,b]$ .
The gradient ∇f = [∂ $f/x$ , ∂ $f/y$ ] exists.

Example: The expanded form a*(∂f/∂x) + b*(∂f/∂y) is equivalent to [a,b]·[∂f/∂x, ∂f/∂y].

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Why rewrite the PDE $5u_x+2u_y=0$ as (5,2)·∇ $u=0$ ?

Rewriting the PDE as a dot product with the gradient allows the equation to be interpreted as a statement about directional derivatives. Specifically, it shows that the directional derivative of u in the direction (5,2) is zero, which implies that u is constant along curves in that direction.

The PDE is $5u_x+2u_y=0$ .
∇u is the gradient vector ( $u_x$ , $u_y$ ).

Example: The lecturer states: 'the first step… is to rewrite it in a particular form. And this will allow us to use arguments involving directional derivatives.' The board shows 'Rewrite PDE as (5,2)·∇ $u = 0$ '.

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What is the relationship between the directional derivative and the gradient vector?

The directional derivative in the direction of a vector w is equal to the dot product of w and the gradient vector ∇f. The gradient vector contains the partial derivatives of the function as its components.

The function $f(x,y)$ is differentiable.
The direction vector is $w=[a,b]$ .
The gradient ∇f = [∂f/∂x, ∂f/∂y] exists.

Example: The directional derivative ∇_w f can be written compactly as w·∇f.

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Does the dot-product formula for directional derivatives work in higher dimensions?

Yes, the dot-product formula w·∇f generalizes naturally to higher dimensions. If the input has five variables, both the direction vector and the gradient vector simply expand to have five components, maintaining the same structural relationship.

The function is differentiable in higher dimensions.
The direction vector and gradient are defined in the same dimensional space.

Example: For a five-dimensional input, the directional derivative is the dot product of a 5-component direction vector and a 5-component gradient vector.

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More concepts to explore

Directional derivatives
A directional derivative measures the local rate of change along a specified direction.
Gradient descent
Iterates move opposite the gradient; step-size and smoothness conditions control convergence.
Integer programming
Force some variables to be whole numbers or binary, which models choosing, counting and sequencing decisions.
Optimization
Derivatives help identify candidate extrema; endpoints, domain restrictions and the distinction between local and global extrema must also be checked.
Partial derivatives
A partial derivative varies one coordinate while holding the others fixed.

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Knowledge connections

Optimization · Related · Map context
Gradient descent · Next concept · Map context
Directional derivatives · Related · Map context
Integer programming · Related · Map context
Total differential · Related · Map context
Partial derivatives · Prerequisite · Map context

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