Explain why or why not Determine whether the following statementsare true and give an explanation or counterexample.a. The rotation field F = ⟨ -y, x⟩ has zero curl and zero divergence.b. ∇ x ∇φ = 0c. Two vector fields with the same curl differ by a constant vector field.d. Two vector fields with the same divergence differ by a constantvector field.e. If F = ⟨x, y, z⟩ and S encloses a region D, then ∫∫S F ⋅ n dS isthree times the volume of D.
Arc Length
Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length.
Line Integral
A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.
Triple Integral
Examples:
Explain why or why not Determine whether the following statements
are true and give an explanation or counterexample.
a. The rotation field F = ⟨ -y, x⟩ has zero curl and zero divergence.
b. ∇ x ∇φ = 0
c. Two vector fields with the same curl differ by a constant vector field.
d. Two vector fields with the same divergence differ by a constant
vector field.
e. If F = ⟨x, y, z⟩ and S encloses a region D, then ∫∫S F ⋅ n dS is
three times the volume of D.
To Check : whether the following statements are true or false:
(a) The rotation field F = ⟨ -y, x⟩ has zero curl and zero divergence.
(b) ∇ x ∇φ = 0.
(c) Two vector fields with the same curl differ by a constant vector field.
(d) Two vector fields with the same divergence differ by a constant.
(e) If F = ⟨x, y, z⟩ and S encloses a region D, then ∫∫S F ⋅ n dS is
three times the volume of D.
Also, give counter examples if statement is false.
(a)
False.
Curl F:
.
Divergence of by F :
.
We see that curlF is , and divergence is 0.
So, the given statement is false.
(b)
True.
=0,
is divergence of some scalar function , thus is some conservative vector function.
Now, is curl of some conservative function.
And we know that curl of a conservative function is 0.
So, ==0.
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