answer:The Industrial Revolution gave rise to several significant political and economic theories. One of the most influential was communism, as outlined by Karl Marx and Friedrich Engels in works such as "The Communist Manifesto." They argued for the abolition of class distinctions and the exploitation of the working class by the bourgeoisie. This ideology led to increased unionization efforts and, ultimately, the Russian Revolution. Another theory that gained prominence was social Darwinism, which applied the concepts of Charles Darwin's evolutionary theory to human societies. It claimed that societies evolved through a natural process of competition, with the strongest and most capable individuals and ideas surviving, while the weak were淘汰. This theory was often used to justify social inequality and laissez-faire economic policies. Additionally, the era saw the development of laissez-faire capitalism, a doctrine that advocated for minimal government intervention in economic affairs, emphasizing individual freedom and the free market. This theory was championed by classical economists like Adam Smith, whose work "The Wealth of Nations" greatly influenced economic thought during the Industrial Revolution.

answer:(a) The given partial derivatives are: F_x = 3z+5y F_y = 3z+5x F_z = 3y+3x To find the potential function f, we antidifferentiate each component with respect to the corresponding variable: f(x,y,z) = int F_x dz = 3xz + 5xy + C_1(y,z) f(x,y,z) = int F_y dx = 3yz + 5xy + C_2(x,z) f(x,y,z) = int F_z dy = 3yz + 3xz + C_3(x,y) Observe that the functions C_1(y,z), C_2(x,z), and C_3(x,y) are already included in the other potential functions. Therefore, the potential function f is: f(x,y,z) = 3xz + 5xy + 3yz + C As f(0,0,0) = 0, we have C = 0, and the final potential function is: boxed{f(x,y,z) = 3xz + 5xy + 3yz} (b) Using the Gradient Theorem, we have: int_C vec F cdot dvec r = int_{(0,0,0)}^{(1,1,1)} nabla f cdot dvec r = f(1,1,1) - f(0,0,0) Evaluating f at the endpoints gives: f(1,1,1) = 3(1)(1) + 5(1)(1) + 3(1)(1) = 3 + 5 + 3 = 11 Hence, boxed{int_C textbf{F} cdot dtextbf{r} = 11}

answer:To find the dot product of two vectors, you multiply the corresponding entries and then sum the results. For vectors A and B, the dot product is calculated as follows: A cdot B = (8)(5) = 40 So, the dot product of the two vectors is 40.

answer:To find the ninth partial sum S_9 of the arithmetic sequence, we first need the common difference d and the term a_9: Given: a_1 = -frac{41}{47} and d = -6sqrt{2} Since each term is found by adding the common difference to the previous term, we have: [a_n = a_{n-1} + d] For n=9, we have: [a_9 = a_1 + (9-1)d] [a_9 = a_1 + 8d] [a_9 = -frac{41}{47} + 8(-6sqrt{2})] [a_9 = -frac{41}{47} - 48sqrt{2}] Now, we apply the formula for the sum of the first n terms: [S_n = frac{n}{2}(2a_1 + (n-1)d)] [S_9 = frac{9}{2}(2(-frac{41}{47}) + (9-1)(-6sqrt{2}))] [S_9 = frac{9}{2}(-frac{82}{47} - 48sqrt{2})] [S_9 = frac{9}{2} left(-frac{82}{47}-48 sqrt{2}right)] The ninth partial sum, S_9, is thus frac{9}{2} left(-frac{82}{47}-48 sqrt{2}right).