Question

explain why or not. Determine whther the ff statements are true or not and give an explaination or counter example

1.The vector field F=<3X^2,1> is a gradient field for both f(x,y)=x^3+y and f(x,y)=y+x^3+100

2.the vector field F=(y,x)/sqrt(x^2+y^2) is constant in direction and magnitude on the unit circle.

3.the vector field F=(Y,X)/SQRT(X^2+Y^2) IS NEITHER RADICAL FIELD NOR A ROTATION FIELD.explain

4.If a curve has a parametric description r(t)=<x(t),y(t),z(t)>, whrer t is the arc length then magnitude of r'(t)=1.explain

5.the vector field F=<y,x> has both zero circulation along and zero flux across the unit circle centered at the origin.explain

6.if F=<y,x> and Cis the circle of radius 4 CENTERED AT (1,0),oriented counterclockwise,then integral F.dr=0.explain tue or false.

7.a paddle wheel with its axis in the direction<0,1,-1 would not spin when put in the vector filed F=<,1,1,2>X<x,y,z..Explain whether True oR False

8.the rotational filed F=<-y,x> has zero curl and zero divergence.tRUE OR fALSE AND eXPLAIN

9.VxVphi=0 .true or false.explain

If F=<X,Y,Z.> and S encloses a region D ,THEN DOUBLE INTEGRAL F.ndS IS THREE TIMES the volume of D.

Answer 1

(1) f₁(x,y) = x³ + y and f₂(x,y) = y + x³ + 100

Gradient field for f₁(x,y):

Gradient field for f₂(x,y):

Here, is called the del operator.

So, the gradient field for both f(x,y) = x³ + y and f(x,y) = y + x³ + 100 is the vector field F = 3x² + = <3x²,1>

The statement is true.

(2) The vector field, F =(y,x)/sqrt(x²+y²)

The magnitude of is 1 (i.e. constant) but the direction is not a constant. Direction of the vector field is a function of x and y.

The statement is false.