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carl123
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Give an example of a formula for a vector field whose graph would closely resemble the one shown. The box for this figure is [−2, 2] x [−2, 2].View attachment 4930
Not sure where to start.
Not sure where to start.
carl123 said:Give an example of a formula for a vector field whose graph would closely resemble the one shown. The box for this figure is [−2, 2] x [−2, 2].
Not sure where to start.
carl123 said:Will it be f (x,y) = f (-2,2)?
I like Serena said:Huh? That's not a formula for f(x,y). (Worried)
Consider that for instance $\frac{1}{x-1}$ is a formula where $x=1$ is a singular point.
carl123 said:where x = 1 is a singular point, the denominator will be 0 then.
I like Serena said:Exactly! (Nod)
Suppose we pick $f(x,y)=\frac{y}{x-1}$.carl123 said:if the denominator is 0, the formula will be undefined. What does that say about the graph?
How would that cross product be related?Also, when I found the cross product of [-2,2] and [-2,2], I got 0. Does that mean, there's no field that resembles the graph?
I like Serena said:Suppose we pick $f(x,y)=\frac{y}{x-1}$.
How would that cross product be related?
[-2,2] x [-2,2] is not the cross product of two vectors. (How can you take the cross product of two vectors in 2D space??) It represents the domain of the function: \(\displaystyle x \in [ -2,2 ] \) and \(\displaystyle y \in [-2,2] \).carl123 said:Also, when I found the cross product of [-2,2] and [-2,2], I got 0. Does that mean, there's no field that resembles the graph?
Thanks.
A vector field is a mathematical concept that describes a vector quantity (such as force, velocity, or electric field) at every point in a given region of space.
A vector field is defined by a set of mathematical functions that assign a vector to each point in a given region of space. These functions can be represented as equations or graphs.
The formula for a vector field depends on the specific type of vector field being described. For example, the formula for a gravitational field would be different from the formula for an electric field. However, in general, a vector field can be represented as F(x,y,z) = (P(x,y,z), Q(x,y,z), R(x,y,z)), where P, Q, and R are the functions that define the x, y, and z components of the vector field, respectively.
A vector field can be graphically represented by drawing arrows (vectors) at different points in the region of space being described. The direction and magnitude of the vector at each point correspond to the direction and magnitude of the vector quantity at that point.
Vector fields have a wide range of applications in physics, engineering, and other fields. Some common examples include describing fluid flow (such as air or water currents), electromagnetic fields, and gravitational fields. They are also used in computer graphics to create realistic simulations of natural phenomena.