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Potential functions Potential functions arise frequently
Chapter 11, Problem 73E(choose chapter or problem)
Potential functions Potential functions arise frequently in physics and engineering. A potential function has the property that a field of interest (for example, an electric field, a gravitational field, or a velocity field) is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 14.)
Velocity potential in two dimensions The motion of an ideal fluid (an incompressible and irrotational fluid) is governed by a velocity potential \(\varphi\). The velocity components of the fluid, u in the x-direction and v in the y-direction, are given by \(\langle u, v)=\nabla \varphi\). Find the velocity components associated with the velocity potential \(\varphi(x, y)=\sin \pi x \sin 2 \pi y .\).
Questions & Answers
QUESTION:
Potential functions Potential functions arise frequently in physics and engineering. A potential function has the property that a field of interest (for example, an electric field, a gravitational field, or a velocity field) is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 14.)
Velocity potential in two dimensions The motion of an ideal fluid (an incompressible and irrotational fluid) is governed by a velocity potential \(\varphi\). The velocity components of the fluid, u in the x-direction and v in the y-direction, are given by \(\langle u, v)=\nabla \varphi\). Find the velocity components associated with the velocity potential \(\varphi(x, y)=\sin \pi x \sin 2 \pi y .\).
ANSWER:Solution 73E
Step 1 of 2:
In this problem we need to find the velocity component associated with the velocity potential
Given:
To find:
The gradient of
We have