So usually for a chain rule, you like to think about canceling the numerator and the denominator. So it's a little bit different than for normal regular derivatives. We can divide through by a dt in the denominator, a delta t, and then if we divide this expression by dt, we get partial of f with respect to x, dx, dt plus partial of f with respect to y, dy, dt, and that's the chain rule for partial derivatives. So we can ask what is the derivative of f with respect to t? So that's the normal single variable derivative. So we want to, f now can be viewed as a multivariable function of x and y or it can be viewed as a single variable function of t, right? So f can be viewed as just a function of t. So how does this help us with the chain rule? Well, let's say for instance f is a function of x which is also a function of t and a function of y which is also a function of t. So you view dx and dy are our h here and the limit h goes to zero, you just view these things as very small, very small numbers like an h, and one is taking the limit in the mind. So that's from the first term, and similarly for the second term, this is just the partial derivative of f with respect to y times dy. So evaluated at y plus dy and y plus d y is the same thing as evaluating our y, and then f of x plus dx minus f of x is just the partial derivative of f with respect to x times this infinitesimal dx. They are small as we would like them to be. But we're considering dx and dy to be small quantities. So this is the same formula for the partial derivative, except holding y fixed at y plus dy. So the definition of the partial derivative is that you hold y fixed and then you take the limit of f of x plus h minus f of x. The minus f of x, y, plus dy cancels the plus f of x, y, plus dy. We can write this as f of x plus dx, y, plus dy minus f of x, y, plus dy plus, I can cancel out this f of x, y, plus dy term by writing this as f of x, y, plus dy and then minus our f of x, y. Can we write that in terms of partial derivatives? We can if we group these terms a little bit differently. So it tells you how f is changing when you change both x and y. That's going to be the value of f and x plus dx, y, plus dy minus the value of f at x and y. I will write down what's called the total differential of f. So let's consider a function f which is a function of two variables only for simplicity. We need to develop a chain rule now using partial derivatives. So everyone knows the chain rule from single variable calculus. There are a total of five weeks to the course, and at the end of each week there is an assessed quiz. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. And after each substantial topic, there is a short practice quiz. The course contains 53 short lecture videos, with a few problems to solve following each lecture. Two semesters of single variable calculus (differentiation and integration) are a prerequisite. Instead of Vector Calculus, some universities might call this course Multivariable or Multivariate Calculus or Calculus 3. These theorems are needed in core engineering subjects such as Electromagnetism and Fluid Mechanics. The fourth week covers line and surface integrals, and the fifth week covers the fundamental theorems of vector calculus, including the gradient theorem, the divergence theorem and Stokes’ theorem. In the first week we learn about scalar and vector fields, in the second week about differentiating fields, in the third week about multidimensional integration and curvilinear coordinate systems. The speedometer reads 60 miles per hour, what is the odometer doing? Besides recording total distance traveled, it is incrementing dutifully every hour by 60 miles.This course covers both the basic theory and applications of Vector Calculus. Imagine motoring along down highway 61 leaving Minnesota on the way to New Orleans though lost in listening to music, still mindful of the speedometer and odometer, both prominently placed on the dashboard of the car. Before defining the derivative of a function, let's begin with two motivating examples.
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