Podcast
Questions and Answers
What is the work done in moving a particle once around a circle of radius 3 in the xy plane under the force field F = (2x - y + z)i + (x + y - z²)j + (3x - 2y + 4z)k?
What is the work done in moving a particle once around a circle of radius 3 in the xy plane under the force field F = (2x - y + z)i + (x + y - z²)j + (3x - 2y + 4z)k?
- 18π
- 9π
- 0 (correct)
- 6π
Which condition must be satisfied for the force field F = (2xy + z³)i + x²j + 3xz²k to be classified as a conservative force field?
Which condition must be satisfied for the force field F = (2xy + z³)i + x²j + 3xz²k to be classified as a conservative force field?
- The divergence of F should be zero.
- The force must depend on time.
- The curl of F should be zero. (correct)
- F must be a function of position alone.
What is the scalar potential for the conservative force field F = (2xy + z³)i + x²j + 3xz²k?
What is the scalar potential for the conservative force field F = (2xy + z³)i + x²j + 3xz²k?
- x²y + 3xz + C (correct)
- 2xyz + x² + C
- xy² + z³ + C
- x³y + z² + C
In finding the area cut from the plane x + 2y + 2z = 5 by the cylinder defined by x = y² and x = 2 - y², which shape do the equations represent?
In finding the area cut from the plane x + 2y + 2z = 5 by the cylinder defined by x = y² and x = 2 - y², which shape do the equations represent?
What is the flux of the field F(x, y, z) = 4xi + 4yj + 2k through the surface created by the intersection of the paraboloid z = x² + y² and the plane z = 1?
What is the flux of the field F(x, y, z) = 4xi + 4yj + 2k through the surface created by the intersection of the paraboloid z = x² + y² and the plane z = 1?
Which equation correctly verifies Green’s theorem in the plane for the vector field (xy + y²)dx + x²dy?
Which equation correctly verifies Green’s theorem in the plane for the vector field (xy + y²)dx + x²dy?
For the closed curve C bounded by y = x and y = x², what would be the essential characteristic of the region it encloses?
For the closed curve C bounded by y = x and y = x², what would be the essential characteristic of the region it encloses?
What is the importance of calculating the work done by a particle in a circular motion under a given force field?
What is the importance of calculating the work done by a particle in a circular motion under a given force field?
Which statement regarding the area found in the intersection of the cylinder and the plane is true?
Which statement regarding the area found in the intersection of the cylinder and the plane is true?
Flashcards
Work done in moving a particle
Work done in moving a particle
The amount of energy required to move a particle along a path under the influence of a force field.
Conservative force field
Conservative force field
A force field where the work done in moving a particle between two points is independent of the path taken.
Scalar potential
Scalar potential
A scalar function whose gradient gives the conservative force field.
Area of a region cut from a plane
Area of a region cut from a plane
The area enclosed by a region of a plane limited by contours from a cylinder.
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Flux of a field
Flux of a field
The rate at which a vector field flows through a surface.
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Green's theorem
Green's theorem
A theorem relating a line integral around a simple closed curve to a double integral over the plane region enclosed by the curve.
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Question 1
- Find the work done moving a particle around a circle C in the xy-plane.
- Circle has center at the origin and radius 3.
- Force field is F = (2x - y + z)i + (x + y - z²)j + (3x - 2y + 4z)k.
Question 2
- Show that F = (2xy + z³)i + x²j + 3xz²k is conservative.
- Find the scalar potential.
Question 3
- Find the area of the region cut from the plane x + 2y + 2z = 5 by the cylinder.
- Cylinder walls are x = y² and x = 2 - y².
Question 4
- Find the flux of the field F(x, y, z) = 4xi + 4yj + 2k outward away from the z-axis.
- Through the surface cut from the bottom of the paraboloid z = x² + y² by the plane z = 1.
Question 5
- Verify Green's Theorem in the plane.
- For ∫(xy + y²)dx + x²dy.
- Where C is the closed curve of the region bounded by y = x and y = x².
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