Answer Since the vector corresponds to the point on the plane, a four-dimensional space is required to draw a graph. Unfortunately, a four-dimensional space cannot be prepared, so the vector field is expressed using the following method. First, select the point on the plane, and then set the vector at that point. Draw the point as the starting point.
If you look at the figure 3.1, you will notice that the vector is a tangent to a curve..This curve is called streamlines or lines of force. Generally, when represents the velocity of a fluid, the curve drawn along the flow is called a streamline, and when represents a magnetic field, it is along the direction of the magnetic field. The drawn curve is called the magnetic field line..Similarly, when represents an electric field, the curve drawn along the direction of the electric field is drawn along the power line, and when represents the electromagnetic field, it is drawn along the direction of the electromagnetic field. The curved line is called an electromagnetic force line.
Bring a magnet to the sandbox and collect iron sand. If you sprinkle this iron sand on paper and place a U magnet under the paper, the iron sand will line up along the lines of magnetic force, and you may have observed that the stronger the magnetic field, the more iron sand is attached. Let us consider these phenomena here..
Electric field
If the distance from the charge to the point P is and the unit vector from to P is , the electric field at the point P is given by the following equation.
Universal gravitational field
for a universal gravitational field (generally called universal gravitational force) in which an object with a substance amount of at the origin acts on an object with a substance amount of at a point P Then
Gradient
Here, for a scalar field defined in an area of space,consider vector field defined by
For the scalar field , the curved surface defined by (c constant) is the level surface of the scalar field . and the group of coordinating surfaces obtained by changing the value of is called the coordinating surface group.
Answer Let the level surface through the point be
Directional derivatives
Let the unit vector be the directional unit vector at point P. Also, a straight line passing through the point P and having as the direction vector is represented by using the distance from the point P. Then, at the point P, the directional derivative of the scalar field in the direction is given by
Therefore,
Answer Let
Next, find the directional unit vector to find the directional derivative in the direction at the point . .Then the directional derivative is
Also, the equation of the tangent plane is
Answer Let be a streamline equation. Then expresses the normal vector of .
Potential Let the position vector of the point P be Then there are many things that are inversely proportional to the distance, such as the magnitude of universal gravitation and the intensity of light. These can be set by
Answer
Answer (1)
(2)
(2) the value of at P.