Question

Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9. $\mathbf{F}(x, y)=\frac{1}{2} x \mathbf{i}+y \mathbf{j}$

   Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9.
$\mathbf{F}(x, y)=\frac{1}{2} x \mathbf{i}+y \mathbf{j}$

Calculus

James Stewart 7th Edition

Chapter 16, Problem 2

Instant Answer

Step 1

The given vector field is \(\mathbf{F}(x, y) = \frac{1}{2} x \mathbf{i} + y \mathbf{j}\). This means that at any point \((x, y)\), the vector has an \(x\)-component of \(\frac{1}{2} x\) and a \(y\)-component of \(y\). Show more…

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Sketch the vector field $\mathbf{F}$ by drawing a diagram like Figure 5 or Figure 9. $\mathbf{F}(x, y)=\frac{1}{2} x \mathbf{i}+y \mathbf{j}$

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Key Concepts

Vector Field

A vector field assigns a vector to every point in a region of space. This concept is used to describe directional quantities, such as velocity or force, that vary with position.

Constant Vector Field

A constant vector field is one in which the vector at every point is identical in both magnitude and direction. This uniformity simplifies analysis and is often represented by parallel arrows of equal length across the entire field.

Graphical Representation

Graphically representing a vector field involves plotting arrows at selected points. Each arrow depicts the vector at that point, with its length indicating magnitude and its angle indicating direction, providing a visual tool for understanding the behavior of the field.

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