Solution Found!
Calculate the surface integral of the function in Ex. 1.7,
Chapter 1, Problem 30P(choose chapter or problem)
Calculate the surface integral of the function in Ex. 1.7, over the bottom of the box. For consistency, let “upward” be the positive direction. Does the surface integral depend only on the boundary line for this function? What is the total flux over the closed surface of the box (including the bottom)? [Note: For the closed surface, the positive direction is “outward,” and hence “down,” for the bottom face.]
Calculate the surface integral of over five sides (excluding the bottom) of the cubical box (side 2) in Fig. 1.23. Let “upward and outward” be the positive direction, as indicated by the arrows.
Figure 1.23
Questions & Answers
QUESTION:
Calculate the surface integral of the function in Ex. 1.7, over the bottom of the box. For consistency, let “upward” be the positive direction. Does the surface integral depend only on the boundary line for this function? What is the total flux over the closed surface of the box (including the bottom)? [Note: For the closed surface, the positive direction is “outward,” and hence “down,” for the bottom face.]
Calculate the surface integral of over five sides (excluding the bottom) of the cubical box (side 2) in Fig. 1.23. Let “upward and outward” be the positive direction, as indicated by the arrows.
Figure 1.23
ANSWER:
Step 1 of 10
Consider a function v and the infinitesimally small area da. The integral over a specified surface S is called as surface integral. The small area da is in the direction perpendicular to the surface. We are going to calculate the surface integral of the given function over the bottom of the box, the relationship between the boundary line and the integral and total flux.
The surface integral is expressed as
The side of the cubical box is 2
The given function is
The small area is expressed as