Drawing a Gradient Field: f(x,y)=xy^2

Thread starter stanford1463
Start date Oct 21, 2008
Tags

Fields Gradient

In summary, the direction of a gradient field is determined by the direction of the arrows pointing in the field, which indicate the steepest increase or decrease in the function. The value of the gradient field represents the rate of change of the function at a given point, with a larger value indicating a steeper change and a smaller value indicating a gentler change. To plot a gradient field, a table of values can be created and arrows can be plotted at each point, or software/online tools can be used. The purpose of drawing a gradient field is to visually represent the rate of change of a function and identify areas of steep increase or decrease. A gradient field can have negative values, indicating a decrease in the function in a particular direction

Oct 21, 2008

stanford1463

Does anyone know how to draw a gradient field? For example, how do you draw one of f(x,y)=xy^2

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Oct 22, 2008

cristo

Staff Emeritus

Science Advisor

Yes.

Oct 22, 2008

RTW69

I believe HallsofIvy answered this question on Feb 11-07 under "How to graph vector fields". must use defination of grad f

1. How do you determine the direction of a gradient field?

The direction of a gradient field is determined by the direction of the arrows pointing in the field. These arrows indicate the direction of the steepest increase or decrease in the function.

2. What does the value of the gradient field represent?

The value of the gradient field represents the rate of change of the function at a given point. A larger value indicates a steeper increase or decrease, while a smaller value indicates a gentler change.

3. How do you plot a gradient field?

To plot a gradient field for a function, you can first create a table of values and then plot arrows at each point, with the direction and length of the arrows determined by the gradient at that point. Alternatively, you can use software or online tools that can generate a gradient field plot for you.

4. What is the purpose of drawing a gradient field?

The purpose of drawing a gradient field is to visually represent the rate of change of a function in different directions. This can help in understanding the behavior of the function and identifying areas of steep increase or decrease.

5. Can a gradient field have negative values?

Yes, a gradient field can have negative values. A negative value indicates a decrease in the function in a particular direction, while a positive value indicates an increase. However, the magnitude of the negative value may not necessarily be greater than the magnitude of the positive value.