Vector Fields and Surface Integrals

This video tutorial covers the second part of surface integrals of a vector field. It begins with a recap of the orientation of surfaces and how it affects the evaluation of surface integrals. The tutorial then provides a detailed example of calculating the flux across a surface using a vector field, with a focus on the upward orientation. The process involves converting to polar coordinates to determine integration limits and performing the necessary calculations to find the flow rate. The video concludes with a summary of the results, emphasizing the positive flow rate due to the upward orientation.

What is the effect of surface orientation on the evaluation of a surface integral?

• It changes the vector field.

• It affects the sign of the Z component.

• It has no effect.

• It determines the limits of integration.

What is the vector field used in the example?

• X, Y, 1

• X, Y, Z

• X, Y, 9 - x^2 - y^2

• X, Y, 0

How is the surface above the XY plane represented graphically?

• As a red surface

• As a yellow surface

• As a blue surface

• As a green surface

What is the partial derivative of G with respect to X?

• 2Y

• -2Y

• 2X

• -2X

What is the Z component of the vector field when the orientation is upward?

• -1

• 0

• 1

• 2

What shape is formed by the XY trace of the surface?

• A square

• A triangle

• A circle

• An ellipse

What is the radius of the circle formed by the XY trace?

• 5

• 4

• 3

• 2

What is the range of R in polar coordinates for this integration?

• 0 to 2

• 0 to 3

• 0 to 4

• 0 to 5

What is the final flow rate across the surface?

• 243 Pi / 8

• 243 Pi / 2

• 243 Pi

• 243 Pi / 4

Why is the flow rate positive in this example?

• Because the vector field is negative.

• Because the vector field is zero.

• Because the surface is oriented downward.

• Because the surface is oriented upward.