Let G be a solid whose surface σ is oriented outward by the unit normal n, and let F x , y , z denote a vector field whose component functions have continuous first partial derivatives on some open set containing G . The Divergence Theorem states that the surface integral ___________ and the triple integral ___________ have the same value.
Let G be a solid whose surface σ is oriented outward by the unit normal n, and let F x , y , z denote a vector field whose component functions have continuous first partial derivatives on some open set containing G . The Divergence Theorem states that the surface integral ___________ and the triple integral ___________ have the same value.
Let G be a solid whose surface
σ
is oriented outward by the unit normal n, and let
F
x
,
y
,
z
denote a vector field whose component functions have continuous first partial derivatives on some open set containing G. The Divergence Theorem states that the surface integral
___________
and the triple integral
___________
have the same value.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Let f be a function of two variables that has continuous partial derivatives and consider the points
A(5, 2), B(13, 2), C(5, 13), and D(14, 14). The directional derivative of f at A in the direction of the vector AB is 4 and the directional derivative at A in the direction of AC is 9.
Find the directional derivative of f at A in the direction of the vector AD.
Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity ∇g · n = Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)
Precalculus: Mathematics for Calculus - 6th Edition
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
01 - What Is an Integral in Calculus? Learn Calculus Integration and how to Solve Integrals.; Author: Math and Science;https://www.youtube.com/watch?v=BHRWArTFgTs;License: Standard YouTube License, CC-BY