Multivariable Calculus (Calc III) - Complete Semester Course - Gradients and Tangent Planes, Multivariable Calculus

The gradient of a scalar-valued function of one variable (z=f(x,y) or w=f(x,y,z)) is orthogonal to level sets of that function. Using this fact to write equations for tangent lines/planes/spaces. ERROR: at 10:09, my second vector in the dot product should be [(x1,x2,...,xn,x(n+1)) - (a1,a2,..,an,f(a1,...,an))]. Multivariable Calculus Unit 3 Lecture 14: We start by considering level set (or level curves for two-dimensional domains) of a scalar-valued function 𝑓. A level set consists of all points in the domain that map to the same specific value in the function's range. For 𝑓(𝑥,𝑦)=𝑥^2+𝑦^2, these curves form concentric circles in the 𝑥𝑦-plane. We then pick a point in the domain where the gradient of the function is non-zero and imagine a parametric curve 𝑟⃗ (𝑡) lying on one of these level curves. By evaluating the function 𝑓 along the parametric curve 𝑟⃗ (𝑡), we find that 𝑓(𝑟⃗ (𝑡)) is a constant function. Differentiating this constant function with respect to 𝑡 using the chain rule, we establish that the gradient of 𝑓 at any point on a level curve is orthogonal (or perpendicular) to the curve at that point: 𝐶=𝑓(𝑟⃗ (𝑡)) 0=𝑑/𝑑𝑡 𝑓(𝑟⃗ (𝑡))=∇𝑓(𝑟⃗ (𝑡))⋅𝑟⃗ ′(𝑡) so ∇𝑓(𝑟⃗ (𝑡))⊥𝑟⃗ ′(𝑡). This observation is crucial as it shows the gradient for a scalar-valued function of two variables is always perpendicular to the level curve it belongs to. The orthogonality of the gradient to the level curves leads to a technique for writing equations of tangent planes to the graph of a function. We introduce a new function ℎ, which reinterprets the graph of 𝑓 as a level set of ℎ; therefore, if 𝑓 : ℝ^𝑛 → ℝ, then ℎ : ℝ^(𝑛+1) → ℝ. The gradient of ℎ is then used to find the orthogonal vector necessary for the equation of the tangent plane for 𝑓. #calculus #multivariablecalculus #mathematics #gradient #partialderivatives #chainrule #tangent #iitjammathematics #calculus3