Keywords [1] The standard \(n\)-dimensional space \(\mathbb{R}^n\) contains all real column vectors with \(n\) components [2] If \(v\) and \(w\) are in a vector space \(S\), every combination...

Directional Derivatives Recall that if \(z = f(x,y)\), then the partial derivatives \(f_x\) and \(f_y\) are [fx(x_0, y_0) = \lim{h \rightarrow 0} \frac{f(x_0 + h, y_0) - f(x_0, y_0)}{h}] [fy...

The Chain Rule Single Variable The chain rule for a single variable gives the rule for differentiating a composite function: If \(y=f(x)\) and \(x=g(t)\), where \(f\) and \(g\) are differenti...

Tangent Planes stewart-calculus-8th-edition Suppose a surface \(S\) has equation \(z=f(x,y)\) where \(f\) has continuous first partial derivativ...

Semantic Segmentation Semantic Segmentation The semantic segmentation task is to label each pixel with a corresponding class. Introduction ...

Keywords [1] \((Ax)^{T} = x^{T}A^{T}\), \((AB)^{T}=B^{T}A^{T}, (A^{-1})^T = (A^T)^{-1}\) [2] The dot product (inner product) is \(x \cdot y = x^Ty\). This is \((1 \times n)(n \times 1) = (1 \...

Keywords [1] Each elimination step \(E_{ij}\) is inverted by \(L_{ij}\). Off the main diagonal change \(-l_{ij}\) to \(+l_{ij}\) [2] The The whole forward elimination process (with no row exc...

Dot Product The dot product or inner product of \(\vec{v}= \begin{pmatrix} v_1 \\\ v_2 \end{pmatrix}\) and \(\vec{w}= \begin{pmatrix} w_1 \\\ w_2 \end{pmatrix}\) is [\vec{v} \cdot \vec{w} = v...

Partial Deriviatives If \(f\) is a function of two variables \(x\) and \(y\), then Partial derivative of \(f\) w.r.t \(x\) at \((a,b)\) is [fx(a,b) = \lim{h \rightarrow 0} \frac{f(a+h, b) - ...

Limits and Continuity Let \(f\) be a function of two variables whose domain \(D\) includes points arbitrarily close \((a,b)\). Then, we say [\lim_{(x,y)\rightarrow (a,b)}f(x,y) = L] if for e...