6.3 Relationship between directional derivatives and the gradient

The gradient vector points in the direction of steepest increase for a function. It's closely tied to directional derivatives, which measure a function's rate of change in any direction. Understanding this relationship is key to grasping how functions behave in multiple dimensions.

Directional derivatives can be calculated using the gradient and a unit vector. The gradient's magnitude equals the maximum directional derivative. This connection helps us visualize a function's behavior and find its steepest ascent or descent directions.

Gradient and Directional Derivatives

Relationship between Gradient and Directional Derivatives

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• The gradient vector $\nabla f(x, y, z)$ is a vector that points in the direction of the maximum rate of change of a function $f(x, y, z)$ at a given point
  • Components of the gradient are the partial derivatives of $f$ with respect to each variable: $\nabla f(x, y, z) = \langle f_x(x, y, z), f_y(x, y, z), f_z(x, y, z) \rangle$
  • The gradient is perpendicular to the level curves or level surfaces of the function at the point where it is calculated
• The directional derivative D_\vec{u}f(x, y, z) measures the rate of change of a function $f(x, y, z)$ in the direction of a unit vector $\vec{u}$ at a given point
  • It can be calculated using the dot product of the gradient and the unit vector: D_\vec{u}f(x, y, z) = \nabla f(x, y, z) \cdot \vec{u}
  • The directional derivative is positive if $\vec{u}$ points in a direction of increasing $f$, negative if $\vec{u}$ points in a direction of decreasing $f$, and zero if $\vec{u}$ is perpendicular to the gradient

Properties of Gradient and Directional Derivatives

• The maximum value of the directional derivative at a point is equal to the magnitude of the gradient vector at that point
  • This maximum occurs when the unit vector $\vec{u}$ is parallel to the gradient vector
  • The maximum directional derivative is denoted as $\|\nabla f(x, y, z)\|$ or \max_{\|\vec{u}\|=1} D_\vec{u}f(x, y, z)
• The minimum value of the directional derivative at a point is the negative of the magnitude of the gradient vector
  • This minimum occurs when the unit vector $\vec{u}$ is antiparallel to the gradient vector
  • The minimum directional derivative is $-\|\nabla f(x, y, z)\|$ or \min_{\|\vec{u}\|=1} D_\vec{u}f(x, y, z)
• The gradient theorem states that if a function $f(x, y, z)$ is differentiable at a point, then the directional derivative exists in every direction at that point and is given by the dot product of the gradient and the unit vector

Differentiability and Chain Rule

Differentiability in Multivariable Functions

• A function $f(x, y, z)$ is differentiable at a point if it is continuous at that point and its partial derivatives exist and are continuous in some neighborhood of the point
  • Differentiability is a stronger condition than continuity, as a function can be continuous but not differentiable (e.g., $f(x, y) = \sqrt{|xy|}$ at $(0, 0)$)
  • If a function is differentiable at a point, then its tangent plane exists at that point, and the gradient vector is normal to the tangent plane
• The linear approximation of a differentiable function $f(x, y, z)$ near a point $(a, b, c)$ is given by: $L(x, y, z) = f(a, b, c) + f_x(a, b, c)(x - a) + f_y(a, b, c)(y - b) + f_z(a, b, c)(z - c)$
  • This approximation is useful for estimating the value of the function near the point and for finding the equation of the tangent plane

Multivariable Chain Rule

• The chain rule for multivariable functions allows us to find the derivative of a composite function, where the input variables are themselves functions of other variables
  • If $z = f(x, y)$ is a differentiable function and $x = g(t)$ and $y = h(t)$ are differentiable functions, then $\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$
  • This can be extended to functions with more than two input variables and more than one intermediate variable
• The multivariable chain rule is useful in applications such as parametric curves and surfaces, where the coordinates are expressed as functions of one or more parameters
  • For example, if a particle's position is given by $\vec{r}(t) = \langle x(t), y(t), z(t) \rangle$, then its velocity is $\vec{v}(t) = \frac{d\vec{r}}{dt} = \langle \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \rangle$