基于Xception边缘检测器生成细粒度图案的矢量图形.

Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the resultant vector or, for short, the resultant.
We use the laws of geometry to construct resultant vectors, followed by trigonometry to find vector magnitudes and directions. For a geometric construction of the sum of two vectors in a plane, we follow the parallelogram rule. Suppose two vectors are at arbitrary positions. Translate either one of...

Vector multiplication of two vectors yields a vector product, with the magnitude equal to the product of the individual vectors multiplied by the sine of the angle between both the vectors and the direction perpendicular to both the individual vectors. As there are always two directions perpendicular to a given plane, one on each side, the direction of the vector product is governed by the right-hand thumb rule.
Consider the cross product of two vectors. Imagine rotating the first vector about...

A plane traveling due north at 180 km/h in still air was found to be 80 km off-course after 30 minutes, deviating approximately 5 degrees east of north. This deviation means the influence of a crosswind alters the plane’s intended trajectory. The actual ground path formed a diagonal, suggesting that the aircraft’s effective ground speed was reduced to 160 km/h and directed slightly to the east due to the wind.By analyzing the displacement from the intended path, the velocity contributed by the...

Every point on a topographical map corresponds to a particular elevation, so the landscape can be modeled as a surface whose height depends on horizontal position. From any given location, a hiker may face infinitely many directions, but only one direction produces the fastest possible increase in elevation. This unique route is called the direction of steepest ascent, and in multivariable calculus, it is represented by the gradient vector of the elevation function.The gradient vector points...

A surface defined by a function of two variables can be understood by examining how it changes along specific directions. When one variable is held constant, the surface reduces to a curve that reflects variation in the other variable. For example, fixing one variable and moving parallel to a coordinate axis produces a cross-sectional curve. The slope of this curve at a given point represents how the function changes in that particular direction, providing a measure of local steepness.By...

A gradient field is a vector field derived from a scalar field. A scalar field assigns a single numerical value to every point in space, such as temperature, pressure, or electric potential. The gradient field describes how that value changes from point to point. It gives both the direction of the fastest increase and the rate of change in that direction.For a scalar field f(x, y), the gradient is written as\begin{equation*}\nabla f=\left\langle \jfrac{\partial f}{\partial x},\jfrac{\partial...