1. The user requests a textbook outline for Higher Mathematics for engineers covering 15 broad topics. 2. Since the request is for a full textbook preparation, which is extensive, I will provide a structured outline for the first topic only as per the guest rule. 3. Topic 1: Second and third order determinants, methods of calculating determinants, basic properties, minors and algebraic complements, concept of n-order determinant. **Problem:** Understand and calculate second and third order determinants and their properties. **Step 1:** Define a determinant of order 2 and 3. - For a 2x2 matrix $$A=\begin{pmatrix}a & b \\ c & d\end{pmatrix}$$, the determinant is $$\det(A) = ad - bc$$. - For a 3x3 matrix $$B=\begin{pmatrix}a & b & c \\ d & e & f \\ g & h & i\end{pmatrix}$$, the determinant is $$\det(B) = a(ei - fh) - b(di - fg) + c(dh - eg)$$. **Step 2:** Explain methods of calculating determinants. - Expansion by minors: Expanding along a row or column using minors and cofactors. - Properties: Determinant changes sign if two rows are swapped, determinant is zero if two rows are equal, determinant of triangular matrix is product of diagonal elements. **Step 3:** Define minors and algebraic complements. - Minor of element $$a_{ij}$$ is the determinant of the matrix formed by deleting the $$i$$th row and $$j$$th column. - Algebraic complement (cofactor) $$A_{ij} = (-1)^{i+j} M_{ij}$$ where $$M_{ij}$$ is the minor. **Step 4:** Concept of n-order determinant. - Determinant of an $$n \times n$$ matrix is defined recursively by expansion along any row or column using minors and cofactors. **Step 5:** Example calculation. Calculate determinant of $$C=\begin{pmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6\end{pmatrix}$$. - Expand along first row: $$\det(C) = 1 \cdot \begin{vmatrix}4 & 5 \\ 0 & 6\end{vmatrix} - 2 \cdot \begin{vmatrix}0 & 5 \\ 1 & 6\end{vmatrix} + 3 \cdot \begin{vmatrix}0 & 4 \\ 1 & 0\end{vmatrix}$$ - Calculate minors: $$= 1(4 \times 6 - 0 \times 5) - 2(0 \times 6 - 1 \times 5) + 3(0 \times 0 - 1 \times 4)$$ $$= 1(24) - 2(-5) + 3(-4) = 24 + 10 - 12 = 22$$ **Final answer:** The determinant of matrix $$C$$ is $$22$$. This completes the explanation and example for the first topic.