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Chapter 14: Q. 16 (page 1095)

How would you show that a given vector field in 鈩�3 is not conservative?

Short Answer

Expert verified

A given vector field in 鈩�3is not conservative when,

F(x,y,z)鈮�鈭�f(x,y,z)鈮�鈭�f鈭�xi+鈭�f鈭�yj+鈭�f鈭�zk

Step by step solution

01

Step 1.Given Information

How would you show that a given vector field in 鈩�3 is not conservative?

02

Step 2. A vector field is conservative 

If a vector field can be described as a gradient, it offers a number of mathematically appealing properties, including the ability to be easily integrated along curves. Such fields are referred to as conservative vector fields.

A vector field that is conservative. F is a vector field that represents the gradient of a function f.

03

Step 3. A conservative vector field F is a vector field that can be written as the gradient of some function f.

That is,

F(x,y,z)=鈭�f(x,y,z)=鈭�f鈭�xi+鈭�f鈭�yj+鈭�f鈭�zk

Hence, we can say that a given vector field in 鈩�3is not conservative, when

F(x,y,z)鈮�鈭�f(x,y,z)鈮�鈭�f鈭�xi+鈭�f鈭�yj+鈭�f鈭�zk

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Most popular questions from this chapter

Find the work done by the vector field

F(x,y)=x3y2i+(y-x)j

in moving an object around the triangle with vertices (1,1),(2,2), and (3,1), starting and ending at (2,2).

Make a chart of all the new notation, definitions, and theorems in this section, including what each new item means in terms you already understand.

F=xln(xz),5z,1y2+1 , where S is the region of the plane with equation 12x鈭�9y+3z=10, where 2鈮�x鈮�3and 5鈮�y鈮�10, with n pointing upwards.

Find the area of S is the portion of the plane with equation y鈭抸 =蟺2

that lies above the rectangle determined by 0 鈮� x 鈮� 4 and 3 鈮� y 鈮� 6.

S is the portion of the saddle surface determined by z = x2 鈭� y2 that lies above and/or below the annulus in the xy-plane determined by the circles with radii

32and2

and centered at the origin.
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