Lecture 61: Introduction to Vector Fields

61.1 Directional Derivative

61.1.1 Basic Concepts of Directional Derivatives

#####Definition: Directionalderivative

description: $$\frac{\partial f}{\partial l}\Bigg|_{(x_0,y_0)}=\lim_{t\to0^+}\frac{f(x_0+t\cos\alpha,y_0+t\cos\beta)-f(x_0,y_0)}t$$

Explanation

• Function:
• The partial derivation with respect to x and to y reflects the change of the function along the x or y direction;
• However, if you need to describe the function changing in other directions, you need to use directional derivatives;
• Concept:
• 1. $\cos \alpha$ and $\cos\beta$ represent the direction of the straight line;
• 2. $x_0+t\cos\alpha,y_0+t\cos\beta$ represents a point in the direction of a straight line;
• 3. $\frac{f(x_0+t\cos\alpha,y_0+t\cos\beta)-f(x_0,y_0)}t$: The direction value of a point is minus the function value and then divided by t. When t->0, the direction derivative of that point is denoted as the directional derivative value;
• Note:
• t can only tend toward 0 upright;

61.1.2 Differentiability Judgment of Directional Derivatives

#####Theorem: Determinationoftheexistenceofdirectionalderivatives

Description: if $z=f(x,y)$ is differentiable, then this point exists in any directional derivative, and the directional derivative equals two partial derivatives, multiplied by their directional cosine, as shown in the following formula:

$$\frac{\partial f}{\partial l}=\frac{\partial f}{\partial x}\cos\alpha+\frac{\partial f}{\partial y}\cos\beta $$

Explanation: Differentiable -> Arbitrarily directional;

61.2 Gradient

#####Definition: Gradient

Description: Suppose $f(x, y)$ has continuity first-order partial derivative at point $P(x_0, y_0)$, then: $$\mathrm{grad}u=f_x(x_0, y_0)\mathbf{i}+f_y(x_0, y_0)\mathbf{j}$$

Explanation: The meaning of gradients

• Each point has infinite directional derivatives, butthe directional derivative along the gradient direction is the largest;
• The maximum value of the directional derivative is themodulus of the gradient;

61.3 Divergence and Curl

#####Definition: Divergence

Description: For a vector field $\mathbf{A}(x,y,z)=(P,Q,R)$:

$$\operatorname{div}\mathbf{A}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$$

Explanation

• Divergence is a number;

#####Definition: Rotateandcross

Description: Given a vector field $A(x,y,z)=({P,Q,R})$, then:

$$\operatorname{curl}\mathbf{A}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\P&Q&R\end{vmatrix}$$

Explanation

• Curl is a vector;