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Chapter 14: Q. 15 (page 1095)

How would you show that a given vector field in 鈩�2 is not conservative?

Short Answer

Expert verified

A given vector field in 鈩�2is not conservative when,F(x,y)鈮�鈭�f(x,y)鈮�鈭�f鈭�xi+鈭�f鈭�yj

Step by step solution

01

Step 1.Given Information

How would you show that a given vector field in 鈩�2is 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)=鈭�f(x,y)=鈭�f鈭�xi+鈭�f鈭�yj

Hence, we can say that a given vector field in 鈩�2is not conservative, when

F(x,y)鈮�鈭�f(x,y)鈮�鈭�f鈭�xi+鈭�f鈭�yj

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Most popular questions from this chapter

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Two different surfaces with the same area. (Try to make these very different, not just shifted copies of each other.)

(b) Let S be the surface parametrized by r(u,z)=x(u,z)i+y(u,v)j+z(u,v)k.

Give two different unit normal vectors to S at the point r(u0,v0).

(c) A smooth surface that can be smoothly parametrized as r(x,z)=x,f(x),z.

Give a formula for a normal vector to the surface S determined by x = f(y, z), where f(y, z) is a function with continuous partial derivatives.

Integrate the given function over the accompanying surface in Exercises 27鈥�34.
f(x,y,z)=xyz2, where S is the portion of the cone 3z=x2+y2 that lies within the sphere of radius 4 and centered at the origin.

ComputethedivergenceofthevectorfieldsinExercises17鈥�22.G(x,y)=xcos(xy),ycos(xy)

What are the outputs of a vector field in the Cartesian plane?
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